On graded nil clean rings

TL;DR

This paper introduces and investigates graded nil clean rings, exploring their properties, extensions to matrix and group rings, and conditions for nil cleanness to be preserved in graded structures.

Contribution

It defines graded nil clean rings, studies their extensions, and examines conditions for nil cleanness to imply graded nil cleanness in various graded ring contexts.

Findings

Characterization of graded nil clean rings

Conditions for nil cleanness to imply graded nil cleanness

Extensions to graded matrix and group rings

Abstract

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.

Equations

$$M_{n}(R)_{\lambda}(\overline{\sigma})=\left(\begin{array}[]{cccc}R_{g_{1}\lambda g_{1}^{-1}}&R_{g_{1}\lambda g_{2}^{-1}}&\dots&R_{g_{1}\lambda g_{n}^{-1}}\\ R_{g_{2}\lambda g_{1}^{-1}}&R_{g_{2}\lambda g_{2}^{-1}}&\dots&R_{g_{2}\lambda g_{n}^{-1}}\\ \vdots&\vdots&\dots&\vdots\\ R_{g_{n}\lambda g_{1}^{-1}}&R_{g_{n}\lambda g_{2}^{-1}}&\dots&R_{g_{n}\lambda g_{n}^{-1}}\\ \end{array}\right).$$

$$R_e\text{is nil clean}\Rightarrow R=\underset{g\in G}{⨁}{R}_g\text{is graded nil clean}.$$

Full text

1 Introduction

Since the introduction of clean rings in [29] as rings in which every element can be written as a sum of an idempotent element and a unit element, many authors have investigated rings in which elements can be written as a sum of an idempotent element and an element with a certain property. Recently, in [6], the notion of a (strongly) nil clean ring was introduced as a ring in which every element can be written as a sum of an idempotent element and a nilpotent element (such that an idempotent element and a nilpotent element mutually commute). Many significant results concerning extensions of such rings to matrix rings have been obtained (see [2, 19, 20]) which are related to the famous Köthe’s Conjecture (see [25]). Also, in [26], an extension to group rings in commutative case is investigated, while in [30] this is generalized to noncommutative case. On the other hand, theory of graded rings has been studied by many authors (see [16, 28]). The aim of this paper is to introduce the graded ring theory into the study of the above mentioned ring element properties. Following [6], we introduce the notion of a graded (strongly) $\mathcal{P}$-clean ring, where $\mathcal{P}$ is a graded ABAB-compatible property of a homogeneous element of a graded ring. In particular, we study graded clean and graded nil clean rings. However, emphasis is on graded (strongly) nil clean rings.

After establishing some characterizations and basic properties of graded (strongly) nil clean rings, we focus on extensions of graded clean and graded nil clean rings to graded matrix rings thus generalizing results from [9] and [6]. As already mentioned, in [30] nil clean group rings are investigated. Here we extend some of the results to graded group rings case. This yields an interesting question on how the graded nil cleanness of a group graded ring depends on the nil cleanness of the component which corresponds to the neutral element of a group. We also take a look at a similar question in the case of rings graded by a partial groupoid (see [15, 18]).

2 Preliminaries

All rings are assumed to be associative with identity. For more details on everything stated in the most of this section, we refer to [16, 28].

Let $R$ be a ring, $G$ a group with the identity element $e,$ and let $\{R_{g}\}_{g\in G}$ be a family of additive subgroups of $R.$ Recall that $R$ is said to be $G$-graded if $R=\bigoplus_{g\in G}R_{g}$ and $R_{g}R_{h}\subseteq R_{gh}$ for all $g,h\in G.$ The set $H=\bigcup_{g\in G}R_{g}$ is called the homogeneous part of $R,$ elements of $H$ are called homogeneous, and subgroups $R_{g}$ $(g\in G)$ are called components. If $a\in R_{g},$ then we say that $a$ has the degree $g.$

In the most of this paper we work in the category whose objects are $G$-graded rings and morphisms are homomorphisms of rings which are degree-preserving.

A right ideal (left, two-sided) $I$ of a graded ring $R=\bigoplus_{g\in G}R_{g}$ is called homogeneous or graded if $I=\bigoplus_{g\in G}I\cap R_{g}.$ If $I$ is a two-sided homogeneous ideal (homogeneous ideal in the rest of the paper), then $R/I$ is a $G$-graded ring with components $(R/I)_{g}=R_{g}/I\cap R_{g}.$ A graded ring $R$ is graded-nil if every homogeneous element of $R$ is nilpotent. We also know that $1\in R_{e}.$

As told in the previous section, we extend some results from [30] to the graded case. In order to do that, we recall a way to make a group ring graded. Let $R=\bigoplus_{g\in G}R_{g}$ be a $G$-graded ring, and observe the group ring $R[G].$ Then, according to [27], $R[G]$ is $G$-graded (actually, strongly graded) with the $g$-th component $(R[G])_{g}=\sum_{h\in G}R_{gh^{-1}}h$ and with the multiplication defined via the rule $(r_{g}g^{\prime})(r_{h}h^{\prime})=r_{g}r_{h}(h^{-1}g^{\prime}hh^{\prime}),$ where $g,g^{\prime},h,h^{\prime}\in G$ and $r_{g}\in R_{g},$ $r_{h}\in R_{h}.$

If $H$ is a normal subgroup of $G,$ then, according to [28], we may observe $R[H]$ as a $G$-graded ring $\bigoplus_{g\in G}(R[H])_{g},$ where $(R[H])_{g}=\bigoplus_{h\in H}R_{gh^{-1}}h,$ and where the multiplication is given by $(r_{g}g^{\prime})(r_{h}h^{\prime})=r_{g}r_{h}(h^{-1}g^{\prime}hh^{\prime}),$ where $g,h\in G,$ $g^{\prime},h^{\prime}\in H$ and $r_{g}\in R_{g},$ $r_{h}\in R_{h}.$

If $R=\bigoplus_{g\in G}R_{g}$ is a $G$-graded ring, then a right $G$-graded $R$-module is a right $R$-module $M$ such that $M=\bigoplus_{x\in G}M_{x},$ where $M_{x}$ are additive subgroups of $M,$ and such that $M_{x}R_{g}\subseteq M_{xg}$ for all $x,g\in G.$ A submodule $N$ of a $G$-graded $R$-module $M=\bigoplus_{x\in G}M_{x}$ is called homogeneous if $N=\bigoplus_{x\in G}N\cap M_{x}.$ The category whose objects are right $G$-graded $R$-modules and morphisms are homomorphisms which are degree-preserving is denoted by $R-gr.$

Let $M=\bigoplus_{x\in G}M_{x}\in R-gr.$ If $\mathrm{END}_{R}(M)_{g}$ consists of endomorphisms $f:M\to M$ such that $f(M_{x})\subseteq M_{gx}$ for all $x\in G,$ then $\mathrm{END}_{R}(M)=\bigoplus_{g\in G}\mathrm{END}_{R}(M)_{g}$ is a $G$-graded ring with respect to the usual addition and multiplication defined by $f\cdot g=f\circ g$ $(f,g\in\mathrm{END}_{R}(M)).$

If $R$ is a $G$-graded ring and $n$ a natural number, then we know that the matrix ring $M_{n}(R)$ over $R$ can be made into a $G$-graded ring in the following way. Let $\overline{\sigma}=(g_{1},\dots,g_{n})\in G^{n},$ $\lambda\in G$ and

[TABLE]

Then $M_{n}(R)=\bigoplus_{\lambda\in G}M_{n}(R)_{\lambda}(\overline{\sigma})$ is a $G$-graded ring with respect to the usual matrix addition and multiplication. Usually, this ring is denoted by $M_{n}(R)(\overline{\sigma}).$

A graded module $M\in R-gr$ is said to be graded simple (or graded irreducible) if $MR\neq 0$ and if the only homogeneous submodules of $M$ are trivial submodules. The graded Jacobson radical $J^{g}(R)$ of a $G$-graded ring $R$ is defined to be the intersection of annihilators of all graded simple graded $R$-modules. It is known that $J^{g}(R)$ coincides with the intersection of all maximal homogeneous right ideals of $R,$ and that it is left-right symmetric. As usual, $J(R)$ denotes the classical Jacobson radical of $R.$

We also recall the notion of a graded ring graded in the following sense.

Definition 2.1** ([18, 16, 15]).**

Let $R$ be a ring, and $S$ a partial groupoid, that is, a set with a partial binary operation. Also, let $\{R_{s}\}_{s\in S}$ be a family of additive subgroups of $R.$ We say that $R=\bigoplus_{s\in S}R_{s}$ is $S$-graded and $R$ induces $S$ (or $R$ is an $S$-graded ring inducing $S$) if the following two conditions hold:

• $i)$

$R_{s}R_{t}\subseteq R_{st}$ whenever $st$ is defined;

• $ii)$

$R_{s}R_{t}\neq 0$ implies that the product $st$ is defined.

We refer to [16] for more information concerning $S$-graded rings inducing $S$ (for taking quotients by homogeneous ideals, one may also consult [14]).

The notion of a graded ring presented in [7], as well as in [3, 8, 22], is equivalent to that from Definition 2.1. There it is studied from the homogeneous point of view by observing the homogeneous part of a graded ring with induced partial addition and everywhere defined multiplication. Such a structure is named anneid [3, 8, 22] and an origin of this approach goes back to [21]. For a survey on anneids, one may also consult [23, 31]. Everything obtained for anneids holds for $S$-graded rings inducing $S$ and vice versa.

The role of a degree-preserving homomorphism is taken by the following notion. If $R$ is an $S$-graded ring inducing $S$ and $R^{\prime}$ is an $S^{\prime}$-graded ring inducing $S^{\prime},$ then a ring homomorphism $f:R\to R^{\prime}$ is called homogeneous [3, 8, 22] if a homogeneous element is mapped to a homogeneous element, and if the fact that $f(x)$ is a nonzero homogeneous element of $R^{\prime}$ implies that $x$ is a homogeneous element of $R.$ The corresponding notion for anneids is simply homomorphism of anneids.

3 Graded nil clean rings

Let $G$ be a group with the identity element $e.$

Definition 3.1**.**

A homogeneous element $r$ of a $G$-graded ring $R$ is called graded nil clean (graded strongly nil clean) if it can be written as a sum of a homogeneous idempotent element $f$ and a homogeneous nilpotent element $b$ (such that $fb=bf$). A $G$-graded ring is called graded nil clean (graded strongly nil clean) if every of its homogeneous elements is graded nil clean (graded strongly nil clean).

Remark 3.2*.*

Let a $G$-graded ring $R=\bigoplus_{g\in G}R_{g}$ be graded (strongly) nil clean. If a homogeneous idempotent is nonzero, it has to be from $R_{e},$ of course. If $x\in R_{g}$ is a nonzero element, then either it is nilpotent or $g=e$ and $x=y+z,$ where $y$ is an idempotent element and $z$ a nilpotent element from $R_{e}$ (such that $yz=zy$), that is, $R_{e}$ is a (strongly) nil clean ring. Obviously, every graded-nil ring is strongly nil clean.

Example 3.3*.*

Let $S$ be a nil clean ring. Then the ring of matrices R=\left(\begin{array}[]{cc}S&S\\ S&S\\ \end{array}\right) is a graded nil clean ring with respect to a well known $\mathbb{Z}$-grading R_{0}=\left(\begin{array}[]{cc}S&0\\ 0&S\\ \end{array}\right), R_{1}=\left(\begin{array}[]{cc}0&S\\ 0&0\\ \end{array}\right), R_{-1}=\left(\begin{array}[]{cc}0&0\\ S&0\\ \end{array}\right), R_{i}=\left(\begin{array}[]{cc}0&0\\ 0&0\\ \end{array}\right) for $i\in\mathbb{Z}\setminus\{-1,0,1\}.$

We may introduce a more general notion following the notion of a clean $\mathcal{P}$-ring from [6], where $\mathcal{P}$ is the so-called ABAB-compatible property that an element of a ring may satisfy. If $R=\bigoplus_{g\in G}R_{g}$ is a $G$-graded ring, and $f$ an idempotent element of $R_{e},$ then by a graded corner ring we mean the $G$-graded ring $fRf=\bigoplus_{g\in G}fR_{g}f.$

Definition 3.4**.**

Let $\mathcal{P}$ be a property that a homogeneous element of a $G$-graded ring $R$ can satisfy. We say that $\mathcal{P}$ is a graded ABAB-compatible property if it satisfies the following conditions:

• $i)$

If $a$ is a homogeneous element of $R$ which has property $\mathcal{P},$ then $-a$ also has property $\mathcal{P};$

• $ii)$

If a homogeneous element $a\in R$ has property $\mathcal{P}$ and $f$ is a homogeneous idempotent element of $R$ such that $af=fa,$ then $faf\in fRf$ has property $\mathcal{P};$

• $iii)$

If $a$ is a homogeneous element of $R$ and $f$ a homogeneous idempotent element of $R$ such that $af=fa,$ and if the elements $faf\in fRf$ and $(1-f)a(1-f)\in(1-f)R(1-f)$ both have property $\mathcal{P}$ as elements of the respective graded corner rings, then $a$ has property $\mathcal{P}$ in $R.$

A $G$-graded ring is said to be graded $\mathcal{P}$-clean if every of its homogeneous elements can be written as a sum of a homogeneous idempotent element and a homogeneous element with graded ABAB-compatible property $\mathcal{P}.$

The following lemma is a graded analogue of Lemma 2.8 in [6]. As in [6], it gives us a characterization of a graded strongly nil clean element since “being a homogeneous nilpotent” is a graded ABAB-compatible property.

Lemma 3.5**.**

Let $R$ be a $G$-graded ring and $M$ a $G$-graded right $R$-module. Also, let $\mathcal{P}$ be a graded ABAB-compatible property. An endomorphism $\varphi\in\mathrm{END}_{R}(M)_{e}$ is then the sum of a homogeneous idempotent element $\epsilon$ and a homogeneous element $\alpha$ which has property $\mathcal{P}$ such that $\alpha\epsilon=\epsilon\alpha$ if and only if there exists a direct sum decomposition $M=A\oplus B,$ where $A,B\in R-gr,$ such that $\varphi|_{A}$ is an element of $\mathrm{END}_{R}(A)$ with property $\mathcal{P}$ and $(1-\varphi)|_{B}$ is an element of $\mathrm{END}_{R}(B)$ with property $\mathcal{P}.$ In case $\varphi\in\mathrm{END}_{R}(M)_{g}$ has property $\mathcal{P},$ where $g\neq e,$ the decomposition of $M$ is trivial.

Proof.

Let $\varphi=\epsilon+\alpha,$ with an idempotent element $\epsilon$ and an element $\alpha$ with property $\mathcal{P},$ where $\epsilon,\alpha\in\mathrm{END}_{R}(M)_{e},$ and also, let us assume that $\epsilon\alpha=\alpha\epsilon.$ If $1$ denotes the identical mapping, then $1-\epsilon\in\mathrm{END}_{R}(M)_{e},$ and let us set $A=(1-\epsilon)(M)$ and $B=\epsilon(M).$ Then $A,B\in R-gr$ and $M=A\oplus B.$ Now, like in the proof of Lemma 2.8 in [6], one may verify that $\varphi|_{A}$ and $(1-\varphi)|_{B}$ have the desired properties.

In the other direction, let $\epsilon$ be the projection onto $B$ with kernel $A.$ Then $\epsilon\in\mathrm{END}_{R}(M)_{e}$ and the rest goes as in the proof of Lemma 2.8 in [6].

The second assertion is obvious. ∎

Corollary 3.6**.**

Let $R$ be a $G$-graded ring and $M\in R-gr.$ An element $\varphi\in\mathrm{END}_{R}(M)_{e}$ is graded strongly nil clean if and only if there exists a direct sum decomposition $M=A\oplus B,$ where $A,B\in R-gr,$ such that $A$ and $B$ are $\varphi$-invariant and such that $\varphi|_{A}\in\mathrm{END}_{R}(A)$ and $(1-\varphi)|_{B}\in\mathrm{END}_{R}(B)$ are nilpotent. In case $\varphi\in\mathrm{END}_{R}(M)_{g}$ is nilpotent, where $g\neq e,$ the decomposition of $M$ is trivial.

Of course, as “being a unit” is ABAB-compatible property (see [6]), “being a homogeneous unit” is a graded ABAB-compatible property. Therefore, we also introduce the following notion.

Definition 3.7**.**

A homogeneous element of a $G$-graded ring $R$ is said to be graded clean if it can be written as a sum of a homogeneous idempotent and a homogeneous unit. A $G$-graded ring is said to be graded clean if every of its homogeneous elements is graded clean.

Remark 3.8*.*

If $R=\bigoplus_{g\in G}R_{g}$ is a graded clean ring, then we obviously have that $R_{e}$ is a clean ring, and that every nonzero homogeneous element not coming from $R_{e}$ is a unit. Also, unlike the classical case (see [6]), a graded nil clean ring does not have to be a graded clean ring. Namely, it is enough to look at Example 3.3. Obviously, every graded division ring, that is, a graded ring in which every homogeneous element is invertible, is graded clean.

Remark 3.9*.*

We believe that this is the appropriate place to put some ideas about some other notions one might want to study. Inspired by the notion of a 2-clean element from [32], if $R$ is a $G$-graded ring, we may define an element $a\in R$ to be graded 2-nil-clean if it can be written as a sum of a homogeneous idempotent element and two homogeneous nilpotent elements. Notice that we do not require a graded 2-nil-clean element to be homogeneous. A $G$-graded ring $R$ is then said to be graded 2-nil-clean if every of its elements is graded 2-nil-clean. As an example, we have a $\mathbb{Z}$-graded ring $R=M_{2}(S),$ with R_{0}=\left(\begin{array}[]{cc}S&0\\ 0&S\\ \end{array}\right), R_{1}=\left(\begin{array}[]{cc}0&S\\ 0&0\\ \end{array}\right), R_{-1}=\left(\begin{array}[]{cc}0&0\\ S&0\\ \end{array}\right), R_{i}=\left(\begin{array}[]{cc}0&0\\ 0&0\\ \end{array}\right) for $i\in\mathbb{Z}\setminus\{-1,0,1\},$ where $S$ is a Boolean ring.

Also, in definition of a graded nil clean element one may also discard the assumption of being a homogeneous element. That would also transfer to the notion of a graded nil clean ring of course. If $S$ is a Boolean ring, and $C_{2}=\{e,g\}$ a cyclic group of order two, one such example is a $C_{2}$-graded ring R=\left(\begin{array}[]{cc}S&S\\ 0&S\\ \end{array}\right), with grading R_{e}=\left(\begin{array}[]{cc}S&0\\ 0&S\\ \end{array}\right), R_{g}=\left(\begin{array}[]{cc}0&S\\ 0&0\\ \end{array}\right).

The next result represents a graded version of Proposition 3.15 in [6].

Lemma 3.10**.**

Let $R=\bigoplus_{g\in G}R_{g}$ be a $G$-graded ring and $I$ a homogeneous ideal of $R$ which is graded-nil. Then $R$ is graded nil clean if and only if $R/I$ is graded nil clean.

Proof.

Suppose that $R$ is graded nil clean. Then $R/I$ is a homomorphic image of $R$ by a degree-preserving homomorphism, and therefore $R/I$ is graded nil clean.

Now, let $R/I$ be a graded nil clean ring. We need to show that $R_{e}$ is a nil clean ring and that every homogeneous element not coming from $R_{e}$ is nilpotent. We follow the proof of Proposition 3.15 in [6] in order to prove that $R_{e}$ is nil clean. So, let us assume that $r\in R_{e}.$ Since $\bar{R}=R/I$ is graded nil clean, $\bar{r}\in(R/I)_{e}=R_{e}/I\cap R_{e}$ can be written as a sum of an idempotent element $\bar{f}\in(R/I)_{e}$ and a nilpotent element $\bar{b}\in(R/I)_{e}.$ Now, if we apply Proposition 27.1 in [1] (see also [24]) to $R_{e},$ we see that a homogeneous idempotent element modulo a homogeneous ideal which is graded-nil can be lifted to a homogeneous idempotent of $R.$ Hence, $\bar{f}$ can be lifted to a homogeneous idempotent $f\in R_{e}.$ Now, $\overline{r-f}$ is a nilpotent element from $(R/I)_{e}.$ Since $I$ is graded-nil, it follows that $r-f$ is a nilpotent element in $R_{e}.$ Therefore, $R_{e}$ is nil clean. Now assume that $r\in R_{g},$ where $g\neq e.$ Then $\bar{r}\in(R/I)_{g}$ is nilpotent since $R/I$ is graded nil clean. According to assumption, $I$ is graded-nil, and so, $r$ is nilpotent, which completes the proof. ∎

It is proved in [6] that a ring $R$ is nil clean if and only if $J(R)$ is nil and $R/J(R)$ is nil clean. We establish a similar structure result for $G$-graded rings.

Lemma 3.11**.**

Let $R=\bigoplus_{g\in G}R_{g}$ be a $G$-graded ring which is graded nil clean. If $G$ is finite, then $J^{g}(R)$ is graded-nil.

Proof.

Since $R$ is graded nil clean, we have that $R_{e}$ is nil clean. According to Proposition 3.16 in [6] applied to the ring $R_{e},$ we have that $J(R_{e})$ is nil. Now, Corollary 4.2 in [4] implies that $J(R_{e})=J^{g}(R)\cap R_{e}.$ Therefore, if $a$ is an element of the $e$-th component of $J^{g}(R),$ it is nilpotent. On the other hand, if $a\in J^{g}(R)_{g},$ where $g\neq e,$ then $a\in R_{g},$ and hence $a$ is nilpotent. Therefore $J^{g}(R)$ is graded-nil. ∎

Corollary 3.12**.**

Let $R$ be a $G$-graded ring and $G$ finite. Then $R$ is graded nil clean if and only if $J^{g}(R)$ is graded-nil and $R/J^{g}(R)$ is graded nil clean.

Remark 3.13*.*

Of course, by Lemma 3.10, if $J^{g}(R)$ is graded-nil and $R/J^{g}(R)$ is graded nil clean, then $R$ is graded nil clean for any cardinality of $G.$

We would also like to have a graded analogue of Corollary 3.22 in [6]. In order to obtain it, let us introduce the notion of a graded strongly $\pi$-regular element and inspect a relationship between such an element and a graded strongly nil clean element.

Definition 3.14**.**

A homogeneous element $a$ of a $G$-graded ring is said to be graded strongly $\pi$-regular if it can be written as a sum of a homogeneous idempotent element $f$ and a homogeneous unit $u$ such that $fa=af$ and $faf$ is nilpotent.

Notice that, as in the classical case, the uniqueness of a graded strongly $\pi$-regular decomposition holds in a $G$-graded ring $R=\bigoplus_{g\in G}R_{g}.$ First assume that a homogeneous element $a$ is a unit. Then the assertion is clear. Now assume that $a\in R_{e}$ and $f+u$ and $f^{\prime}+v$ are both strongly $\pi$-regular decompositions of $a.$ It clearly suffices to show that $f=f^{\prime}.$ However, this follows directly from Proposition 2.6 in [6] applied to $R_{e}.$

Of course, every graded strongly nil clean element is strongly nil clean element and hence a strongly $\pi$-regular element according to Proposition 3.5 in [6].

Lemma 3.15**.**

Let $R=\bigoplus_{g\in G}R_{g}$ be a $G$-graded ring and let $a\in R$ be a graded strongly $\pi$-regular element with graded strongly $\pi$-regular decomposition $a=f+u.$ Then $a$ is graded strongly nil clean element if and only if $2f-1+u$ is nilpotent and $u\in R_{e}.$

Proof.

Let $a$ be a graded strongly nil clean element. If $0\neq a\in R_{g},$ where $g\neq e,$ then $f=0$ and $u=a$ is a unit. Since $a$ is by assumption graded strongly nil clean element, we have that $u$ is also a nilpotent element, which is impossible. Therefore, $a\in R_{e}.$ Then $a=f^{\prime}+b$ for some idempotent element $f^{\prime}\in R_{e}$ and a nilpotent element $b\in R_{e}$ which commutes with $f^{\prime}.$ Also, $a=(1-f^{\prime})+(2f^{\prime}-1+b)$ is a strongly $\pi$-regular decomposition of $a$ in $R_{e}$ (see Proposition 3.5 in [6]). Since strongly $\pi$-regular decomposition is unique according to Proposition 2.6 in [6], we have that $f=1-f^{\prime}$ and $u=2f^{\prime}-1+b.$ However, we then have that $2f-1+u=b$ is a nilpotent element, and, of course, $u\in R_{e}.$

The converse follows from Proposition 3.9 in [6] applied to the ring $R_{e}.$ ∎

With this in mind, a graded version of Theorem 3.21 in [6] can be proved which implies a graded version of Corollary 3.22 in [6].

Theorem 3.16**.**

Let $R=\bigoplus_{g\in G}R_{g}$ be a $G$-graded ring and $I$ a homogeneous nilpotent ideal of $R.$ If $a$ is a homogeneous element of $R$ such that $\bar{a}$ is a graded strongly nil clean element in $\bar{R}=R/I,$ then $a$ is a graded strongly nil clean element in $R.$

Proof.

First assume that $\bar{a}\in(R/I)_{e}.$ Since $\bar{a}$ is a graded strongly nil clean element in $R/I,$ it can be written as a sum of an idempotent element $\bar{f}\in(R/I)_{e}$ and a nilpotent element $\bar{b}\in(R/I)_{e}$ such that $\bar{f}\,\bar{b}=\bar{b}\bar{f}.$ Every strongly nil clean element is strongly $\pi$-regular, and $\overline{1-f}+\overline{2f-1+b}$ is a strongly $\pi$-regular decomposition of $\bar{a}$ in $(R/I)_{e}.$ Following the proof of Theorem 3.21 in [6], there exists an idempotent $f^{\prime}\in R_{e}$ and a unit $u\in R_{e}$ such that $a=f^{\prime}+u$ is a strongly $\pi$-regular decomposition of $a$ in $R_{e}.$ It is enough to show that $2f^{\prime}-1+u$ is nilpotent by Lemma 3.15. By using equalities $\bar{f^{\prime}}=\overline{1-f}$ and $\bar{u}=\overline{2f-1+b}$ one gets $\overline{2f^{\prime}-1+u}=\bar{b}.$ Then $2f^{\prime}-1+u$ is nilpotent by the nilpotency of $\bar{b}$ and $I.$ Now assume that $\bar{a}\in(R/I)_{g},$ where $g\neq e.$ Then $\bar{a}$ is a nilpotent element. But the assumption on $I$ implies that $a$ is nilpotent in $R,$ which completes the proof. ∎

Corollary 3.17**.**

Let $R$ be a $G$-graded ring and $I$ a homogeneous nilpotent ideal of $R.$ Then $R$ is graded strongly nil clean if and only if $R/I$ is graded strongly nil clean.

We now turn our attention to matrix rings.

In [9] it is proved that a matrix ring over a clean ring is also clean. We do not have such a result in the $G$-graded setting. However, we have a graded version of Theorem in [9].

Theorem 3.18**.**

Let $R=\bigoplus_{g\in G}R_{g}$ be a $G$-graded ring and let $1=f_{1}+\dots+f_{n}$ in $R,$ where the $f_{i}$ are orthogonal idempotents from $R_{e}.$ If each $f_{i}Rf_{i}$ is graded clean, and if $R$ has no nonzero homogeneous zero divisors, then $R$ is graded clean.

We first prove the following lemma which represents a graded version of Lemma in [9].

Lemma 3.19**.**

Let $f$ be a homogeneous idempotent of a $G$-graded ring $R=\bigoplus_{g\in G}R_{g}$ and let $\bar{f}=1-f.$ Let us assume that $fRf$ and $\bar{f}R\bar{f}$ are graded clean rings. If $R$ has no nonzero homogeneous zero divisors, then $R$ is a graded clean ring.

Proof.

We observe $R$ in its Peirce decomposition R=\left(\begin{array}[]{cc}fRf&fR\bar{f}\\ \bar{f}Rf&\bar{f}R\bar{f}\\ \end{array}\right). According to Lemma in [9], $R_{e}$ is clean. Now, let $0\neq A\in R_{g},$ where $g\neq e,$ and let A=\left(\begin{array}[]{cc}a&x\\ y&b\\ \end{array}\right). Then $a,x,y,b\in R_{g}$ and $a,b$ are units. Let $a_{1}\in R_{g^{-1}}$ be an inverse of $a.$ Then $b-ya_{1}x\in\bar{f}R\bar{f},$ and $b-ya_{1}x\in R_{g}.$ Hence, if $b\neq ya_{1}x,$ we have that $b-ya_{1}x=v$ is a unit. Therefore \left(\begin{array}[]{cc}a&x\\ y&b\\ \end{array}\right)=\left(\begin{array}[]{cc}a&x\\ y&v+ya_{1}x\\ \end{array}\right) is a unit, as it can be shown as in Lemma in [9] for the classical case, since $ya_{1}$ and $a_{1}x$ both belong to $R_{e}.$ Namely, \left(\begin{array}[]{cc}f&0\\ -ya_{1}&\bar{f}\\ \end{array}\right), \left(\begin{array}[]{cc}f&-a_{1}x\\ 0&\bar{f}\\ \end{array}\right) and \left(\begin{array}[]{cc}a&0\\ 0&v\\ \end{array}\right) are units, and, on the other hand, we have \left(\begin{array}[]{cc}f&0\\ -ya_{1}&\bar{f}\\ \end{array}\right)\left(\begin{array}[]{cc}a&x\\ y&v+ya_{1}x\\ \end{array}\right)\left(\begin{array}[]{cc}f&-a_{1}x\\ 0&\bar{f}\\ \end{array}\right)=\left(\begin{array}[]{cc}a&0\\ 0&v\\ \end{array}\right). The case $v=0$ cannot occur since by assumption $R$ has no nonzero homogeneous zero divisors. ∎

Proof of Theorem 3.18.

The assertion follows from the previous lemma by using mathematical induction just as in the case of Theorem in [9]. ∎

When it comes to graded nil clean rings, we establish a graded analogue of an extension of nil clean rings to triangular matrix rings from [6].

Theorem 3.20**.**

Let $R$ be a $G$-graded ring and $n$ a natural number. Then $R$ is graded (strongly) nil clean if and only if $T_{n}(R)(\overline{\sigma})$ is graded (strongly) nil clean triangular matrix ring for every $\overline{\sigma}\in G^{n}.$

Proof.

Let $\overline{\sigma}\in G^{n}$ and $S=T_{n}(R)(\overline{\sigma}).$ Like in the proof of Theorem 4.1 in [6], let us observe the ideal $I$ of $S$ which consists of matrices of $S$ with zeroes along the main diagonal. Ideal $I$ is a homogeneous nilpotent ideal of $S$ and, as in the classical case, it can be proved that $S/I$ is isomorphic to the direct product of $n$ copies of $R.$ Since $I$ is in particular graded-nil, the assertion for graded nil cleanness follows from Lemma 3.10 and the fact that a product of finitely many graded nil clean rings is again a graded nil clean ring (category of graded rings is closed for finite products according to [28]). The statement for graded strongly nil cleanness follows from Corollary 3.17 and the fact that a product of finitely many graded strongly nil clean rings is again a graded strongly nil clean ring. ∎

In [30], the nil cleanness of group rings is investigated. We would like to do similar in group graded setting.

We start with a simple lemma which is also of an independent interest.

Lemma 3.21**.**

If a $G$-graded ring $R=\bigoplus_{g\in G}R_{g}$ is a graded nil clean ring, then $2$ is nilpotent.

Proof.

Since $R$ is a graded nil clean ring, we have that $R_{e}$ is nil clean. Since $1\in R_{e},$ the statement follows directly from Proposition 3.14 in [6]. ∎

As is known, if $G$ is a group, and $H$ a normal subgroup of $G,$ then a $G$-graded ring $R=\bigoplus_{g\in G}R_{g}$ can be viewed as a $G/H$-graded ring with respect to grading $\bigoplus_{C\in G/H}R_{C},$ where $R_{C}=\bigoplus_{x\in C}R_{x}.$

Theorem 3.22**.**

Let $G$ be a locally $2$-finite group and $H$ a normal subgroup of $G.$ Also, assume that $R=\bigoplus_{g\in G}R_{g}$ is a $G$-graded ring which is graded nil clean as a $G/H$-graded ring. Then the $G/H$-graded group ring $R[H]$ is graded nil clean.

Proof.

Following the proof of Theorem 2.3 in [30], we may assume that $H$ is a finite $2$-group. According to [28], page 180, the augmentation mapping $R[H]\to R,$ given by $\sum_{h\in H}r^{h}h\mapsto\sum_{h\in H}r^{h},$ where $R[H]$ is considered as a $G/H$-graded ring, is degree-preserving. Therefore, the kernel of the augmentation mapping, that is, the augmentation ideal $\Delta(R[H]),$ is homogeneous. This means that $R[H]/\Delta(R[H])$ is a $G/H$-graded ring. Moreover, $R[H]/\Delta(R[H])$ and $R$ are isomorphic as $G/H$-graded rings. Hence $R[H]/\Delta(R[H])$ is graded nil clean. Now, $2$ is nilpotent by Lemma 3.21. Theorem 9 in [5] tells us that $\Delta(R[H])$ is nilpotent, and therefore, graded nil. Applying Lemma 3.10 completes the proof. ∎

Theorem 3.23**.**

Let $R=\bigoplus_{g\in G}R_{g}$ be a $G$-graded ring which has only homogeneous idempotent and nilpotent elements. If $R[G]$ is graded nil clean, then $R$ is graded nil clean.

Proof.

Since $R[G]$ is graded nil clean, $(R[G])_{e}$ is nil clean. According to Proposition 2.1(4) in [27], the mapping $f:R\to(R[G])_{e},$ $f(\sum_{g\in G}r_{g})=\sum_{g\in G}r_{g}g^{-1},$ is a ring isomorphism. Therefore, $R$ is nil clean and hence graded nil clean. ∎

Remark 3.24*.*

If $R=\bigoplus_{g\in G}R_{g}$ is a $G$-graded ring such that $R[G]$ has only homogeneous idempotent and nilpotent elements, $G$ a locally 2-finite group, and $(R[G])_{e}$ is nil clean, then $R[G]$ is a graded nil clean ring. Namely, this is a corollary to Theorem 2.3 in [30] since $(R[G])_{e}\cong R.$

Previous remark yields an interesting question of what can be said of the following implication:

[TABLE]

The following example proves that the above implication does not hold in general.

Example 3.25*.*

Let $S$ be a Boolean ring, $G=\{e,g\}$ a cyclic group of order $2,$ and R=\left(\begin{array}[]{cc}S&S\\ S&S\\ \end{array}\right). Then R=\left(\begin{array}[]{cc}S&0\\ 0&S\\ \end{array}\right)\oplus\left(\begin{array}[]{cc}0&S\\ S&0\\ \end{array}\right) is a $G$-graded ring whose $e$-th component R_{e}=\left(\begin{array}[]{cc}S&0\\ 0&S\\ \end{array}\right) is a nil clean ring, but $R$ is not a graded nil clean ring since not all elements of R_{g}=\left(\begin{array}[]{cc}0&S\\ S&0\\ \end{array}\right) are nilpotent.

We continue by giving some sufficient conditions for the above implication to be true.

Theorem 3.26**.**

Let $R=\bigoplus_{g\in G}R_{g}$ be a $G$-graded PI ring which is Jacobson radical. Then, if $R_{e}$ is nil clean, $R$ is graded nil clean.

Proof.

According to Proposition 3.16 in [6], if $S$ is a nil clean ring, then $J(S)$ is nil. Therefore, our assumption yields that $J(R_{e})$ is nil. Now, Theorem 3 in [17] tells us that $J(R)$ is nil since $R$ is by assumption PI. However, $R$ is by assumption Jacobson radical ring. Therefore $R=J(R)$ is a nil ring. In particular, every homogeneous element is nilpotent. Hence $R$ is a graded nil clean ring. ∎

Theorem 3.27**.**

Let $R=\bigoplus_{g\in G}R_{g}$ be a $G$-graded PI ring which is graded local, that is, it has a unique maximal homogeneous right ideal, and let $G$ be a finite group such that the order of $G$ is a unit in $R.$ If $R_{g}R_{g^{-1}}=0$ for every $g\in G\setminus\{e\},$ then, if $R_{e}$ is nil clean, $R$ is graded nil clean.

Proof.

We know from Corollary 3.17 in [6] that a ring $A$ is nil clean if and only if $J(A)$ is nil and $A/J(A)$ is nil clean. Therefore, our assumption yields that $J(R_{e})$ is nil and that $R_{e}/J(R_{e})$ is nil clean. $R$ is by assumption PI, and hence, Theorem 3 in [17] implies that $J(R)$ is nil. Since $G$ is finite, we have that $J(R_{e})=J^{g}(R)\cap R_{e},$ according to Corollary 4.2 in [4]. Also, our assumption on the order of $G$ implies that $J(R)$ is homogeneous and $J^{g}(R)=J(R)$ (see Theorem 4.4 in [4]). Therefore, $R/J(R)$ is a $G$-graded ring. Since $R$ is a graded local ring, we have that $R/J(R)$ is a graded division ring. If $H_{R}$ is the homogeneous part of $R,$ let $H_{R/J(R)}=\bigcup_{g\in G}R_{g}/(J(R)\cap R_{g})=H_{R}/J(R)\cap H_{R}$ be the homogeneous part of $R/J(R)$ with induced partial addition and everywhere defined multiplication, that is, the corresponding anneid. Then $H_{R/J(R)}$ is a simple anneid, that is, it has no nontrivial ideals. Let $f:H_{R/J(R)}\to R_{e}/J(R_{e})$ be the mapping defined by $f(x+J(R)\cap H_{R})=x+J(R_{e})$ if $x\in R_{e}$ and $f(x+J(R)\cap H_{R})=0+J(R_{e})$ if $x\notin R_{e}$ or $x\in J(R),$ and where $x+J(R)\cap H_{R}\in H_{R/J(R)}$ (see also the proof of Theorem 3.2 in [13]). It is easily seen that $f$ is well defined and that it is a surjective homomorphism of anneids. Also, since $J(R)$ is proper and $1\in R_{e},$ we have that $R_{e}\neq J(R_{e}).$ Therefore $\ker f=0.$ Hence $H_{R/J(R)}\cong R_{e}/J(R_{e}).$ It follows that every homogeneous element from $R/J(R)$ is graded nil clean. Hence, $R/J(R)$ is graded nil clean. Finally, according to Lemma 3.10, $R$ is graded nil clean. ∎

We conclude this paper by observing the implication (3.1) in the case of rings graded in the sense of Definition 2.1.

Of course, while observing (3.1) in the case of $S$-graded rings inducing $S,$ letter $e$ would stand for an idempotent element of $S.$ Definition of a graded nil clean element of an $S$-graded ring inducing $S$ as well as of a graded nil clean ring is the same as in the case of a group graded ring. However, $S$ may have more than one nonzero idempotent. Consequently, components of a graded nil clean ring corresponding to these idempotents are all nil clean rings. Of course, here we also have that homogeneous elements, not belonging to components which correspond to nonzero idempotent elements of $S,$ are nilpotent.

In the proof of the next theorem we use notions of the graded Jacobson $J^{g}(R)$ and the large graded Jacobson radical $J^{g}_{l}(R)$ of an $S$-graded ring $R$ inducing $S,$ which are introduced in [7]. We do not recall these notions here since we only need their properties. For more information on these and related radicals, one may also consult [11, 12, 13].

In what follows, we assume that all $S$-graded rings inducing $S$ have an identity. If $S$ is cancellative, then, according to [8], the number of nonzero idempotents of $S$ is finite, all components corresponding to these idempotents have an identity, and an identity of the whole ring is a sum of identities of the aforementioned components.

With this in mind, we record the following characterization of $S$-graded nil clean rings inducing $S.$

Lemma 3.28**.**

Let $S$ be a cancellative partial groupoid, $R=\bigoplus_{s\in S}R_{s}$ an $S$-graded ring inducing $S$ and $I$ a homogeneous ideal of $R$ which is graded-nil. Then $R$ is graded nil clean if and only if $R/I$ is graded nil clean.

Proof.

If $R$ is graded nil clean then $R/I$ is graded nil clean since it is a homomorphic image of $R$ taken by a homogeneous homomorphism.

Now, let $R/I$ be a graded nil clean ring. Let $r$ be a homogeneous element of $R.$ Assume first that $r\in R_{e},$ where $e$ is an arbitrary nonzero idempotent element of $S.$ As in the proof of Lemma 3.10, one obtains that $R_{e}$ is nil clean. Again the case of $r\in R_{g},$ where $g\neq e,$ is dealt easily just as in the proof of Lemma 3.10. ∎

Theorem 3.29**.**

Let $S$ be a finite cancellative partial groupoid, and $R=\bigoplus_{s\in S}R_{s}$ an $S$-graded ring inducing $S$ which is also PI and Jacobson radical ring. If $S$ has exactly one nonzero idempotent element $e,$ and if $R_{e}$ is nil clean, then $R$ is graded nil clean.

Proof.

We first note that $J(R_{e})$ is nil according to Proposition 3.16 in [6]. Since $R$ is Jacobson radical ring, we have that $J(R)$ is homogeneous. Therefore $J(R)$ coincides with the large graded Jacobson radical $J^{g}_{l}(R).$ Namely, according to [7], the largest homogeneous ideal of $R$ contained in $J(R)$ coincides with $J^{g}_{l}(R).$ On the other hand, $J^{g}_{l}(R)\subseteq J^{g}(R)$ (see [7]), which implies that $J^{g}(R)=J^{g}_{l}(R)=R.$ Now, since all the assumptions of Theorem 12 in [10] are satisfied, we have that $J(R)$ is nil. Therefore $R=J(R)$ is nil and hence graded nil clean. ∎

Theorem 3.30**.**

Let $S$ be a finite cancellative partial groupoid, $F$ a field with $\mathrm{char}(F)=0$ or $\mathrm{char}(F)>|S|,$ and let $R=\bigoplus_{s\in S}R_{s}$ be an $S$-graded $F$-algebra inducing $S$ which is also PI. Assume also that $R$ is graded local ring, that is, that it has a unique maximal homogeneous right ideal. If $S$ has exactly one nonzero idempotent element $e$ such that $st=e\Rightarrow s=e\vee t=e$ $(s,t\in S),$ if $J^{g}(R)=J^{g}_{l}(R),$ and if $R_{e}$ is nil clean, then $R$ is graded nil clean.

Proof.

We first note that $J(R_{e})$ is nil and $R_{e}/J(R_{e})$ is nil clean according to Corollary 3.17 in [6]. Now, $J(R)$ is nil according to Theorem 12 in [10]. Also, $J(R)$ is homogeneous by Corollary 4 in [18]. Therefore $J(R)=J^{g}(R)$ since $J^{g}_{l}(R)$ coincides with the largest homogeneous ideal of $R$ contained in $J(R)$ (see [7]). Hence $R/J(R)$ is an $S$-graded ring inducing $S.$ Also, since $J(R_{e})=J^{g}(R)\cap R_{e}$ holds, according to [7], the $e$-th component of $R/J(R)$ is $R_{e}/J(R_{e}).$ The rest of the proof goes as in the proof of Theorem 3.27 with the help of Lemma 3.28. ∎

Acknowledgements

The authors would like to express their sincere gratitude to Professor Ivan Shestakov and to the referee for handling this manuscript. The second author was supported by TUBITAK (No. 117F070).