Prime Structures in a Morita Context

TL;DR

This paper investigates the conditions under which a Morita context related to rings and modules is prime or semiprime, providing necessary and sufficient criteria for these properties.

Contribution

It offers new necessary and sufficient conditions for the primeness and semiprimeness of ideals within Morita contexts, advancing understanding in ring theory.

Findings

Characterization of prime ideals in Morita contexts

Criteria for semiprime ideals in Morita contexts

Conditions for Morita contexts to be prime or semiprime

Abstract

In this paper, we study on the primeness and semiprimeness of a Morita context related to the rings and modules. Necessary and sufficient conditions are investigated for an ideal of a Morita context to be a prime ideal and a semiprime ideal. In particular, we determine the conditions under which a Morita context is prime and semiprime.

Equations

$$\left(\begin{array}[]{cc}r_{1}&v_{1}\\ w_{1}&s_{1}\\ \end{array}\right)\left(\begin{array}[]{cc}r_{2}&v_{2}\\ w_{2}&s_{2}\\ \end{array}\right)=\left(\begin{array}[]{cc}r_{1}r_{2}+sv_{1}w_{2}&r_{1}v_{2}+v_{1}s_{2}\\ w_{1}r_{2}+s_{1}w_{2}&sw_{1}v_{2}+s_{1}s_{2}\\ \end{array}\right).$$

$$\left(\begin{array}[]{cc}0&x\\ 0&0\\ \end{array}\right)\left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right)\left(\begin{array}[]{cc}0&x\\ 0&0\\ \end{array}\right)=\left(\begin{array}[]{cc}0&xWx\\ 0&0\\ \end{array}\right)\subseteq\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right).$$

$$\left(\begin{array}[]{cc}r_{1}&v_{1}\\ w_{1}&s_{1}\\ \end{array}\right)\left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right)\left(\begin{array}[]{cc}r_{2}&v_{2}\\ w_{2}&s_{2}\\ \end{array}\right)\subseteq\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right).$$

$$\left(\begin{array}[]{cc}r_{1}Rr_{2}+v_{1}Wr_{2}+r_{1}Vw_{2}+v_{1}Sw_{2}&r_{1}Rv_{2}+v_{1}Wv_{2}+r_{1}Vs_{2}+v_{1}Ss_{2}\\ w_{1}Rr_{2}+s_{1}Wr_{2}+w_{1}Vw_{2}+s_{1}Sw_{2}&w_{1}Rv_{2}+s_{1}Wv_{2}+w_{1}Vs_{2}+s_{1}Ss_{2}\\ \end{array}\right)$$

$$\left(\begin{array}[]{cc}\overline{3}&\overline{0}\\ \overline{0}&\overline{3}\\ \end{array}\right)\left(\begin{array}[]{cc}\mathbb{Z}_{6}&\overline{2}\mathbb{Z}_{6}\\ \overline{3}\mathbb{Z}_{6}&\mathbb{Z}_{6}\\ \end{array}\right)\left(\begin{array}[]{cc}\overline{1}&\overline{0}\\ \overline{0}&\overline{2}\\ \end{array}\right)=\left(\begin{array}[]{cc}\overline{3}\mathbb{Z}_{6}&0\\ \overline{3}\mathbb{Z}_{6}&0\\ \end{array}\right)\subseteq H,$$

$$(v+V_0)(w+W_0)=vw+P(R),(w+W_0)(v+V_0)=wv+P(S)$$

$$T/P(T)\cong\left(\begin{array}[]{cc}R/P(R)&V/V_{0}\\ W/W_{0}&S/P(S)\\ \end{array}\right).$$

Full text

2010 MSC: 16D80, 16S50, 16S99

Key words: Morita context, prime ideal, semiprime ideal, prime radical, prime ring, semiprime ring

1. Introduction

Throughout this paper all rings are associative with $1\neq 0$ and modules are unitary. A proper ideal $I$ of a ring $R$ is called prime if for any elements $a$ and $b$ in $R$, $aRb\subseteq I$ implies $a\in I$ or $b\in I$, equivalently if for every pair of ideals $A,B$ of $R$, $AB\subseteq I$ implies $A\subseteq I$ or $B\subseteq I$. The prime radical of a ring $R$ is the intersection of all prime ideals of $R$ and denoted by $P(R)$. It is clear that $P(R)$ contains every nilpotent ideal of $R$. Recall that a proper ideal $I$ of $R$ is said to be semiprime if for any $a\in R$, $aRa\subseteq I$ implies $a\in I$, equivalently for every ideal $A$ of $R$, $A^{2}\subseteq I$ implies $A\subseteq I$. The concepts of prime and semiprime ideals have very important roles in the ring theory. A ring $R$ is called prime if [math] is a prime ideal, equivalently for any $a,b\in R$, $aRb=0$ implies $a=0$ or $b=0$. A ring $R$ is said to be semiprime if it has no nonzero nilpotent ideal, equivalently for any $a\in R$, $aRa=0$ implies $a=0$. Obviously, prime rings are semiprime. Also, it is well-known that a ring $R$ is semiprime if and only if $P(R)=0$. A proper submodule $N$ of a left $R$-module $M$ is called prime if for any $r\in R$ and $m\in M$, $rRm\subseteq N$ implies $rM\subseteq N$ or $m\in N$. It is easy to show that if $N$ is a prime submodule of $M$, then the annihilator of the module $M/N$ is a two-sided prime ideal of $R$. A module $M$ is said to be prime if [math] is a prime submodule of $M$, i.e., for any $r\in R$ and $m\in M$, $rRm=0$ implies $rM=0$ or $m=0$.

A Morita context is a 4-tuple $(R,V,W,S)$, where $R,S$ are rings, ${}_{R}V_{S}$ and ${}_{S}W_{R}$ are bimodules with context products $V\times W\rightarrow R$ and $W\times V\rightarrow S$ written multiplicatively as $(v,w)\mapsto vw$ and $(w,v)\mapsto wv$, such that \left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right) is an associative ring with the usual matrix operations. Morita contexts appeared in the work of Morita [5] and they play an important role in the category of rings and modules that described equivalences between full categories of modules over rings. One of the fundamental results in this direction says that the categories of left modules over the rings $R$ and $S$ are equivalent if and only if there exists a strict Morita context connecting $R$ and $S$. This concept appears in disguise in the literature of the ring theory. A Morita context forms a very large class of rings generalizing matrix rings. Formal triangular matrix rings are some of the obvious examples of Morita contexts, and have been studied extensively. The concept of Morita context has been studied by many researchers (see, e.g.,[2, 3, 4, 7]). In [6], Sands investigated radicals of Morita context and obtained valuable results about general cases of radicals. Recently, in [7], Tang et. al. specify in detail about the structure of ideals in a Morita context, worthy results are exhibited concerned with the structures of some special rings $K_{s}(R)$ and some cases of a Morita context. $K_{s}(R)$ is a special kind of the Morita context. Let $R=V=W=S$ and $s$ is a central element of $R$. The 4-tuple \left(\begin{array}[]{cc}R&R\\ R&R\\ \end{array}\right) becomes a ring with usual matrix addition and with multiplication defined by

[TABLE]

This ring is denoted by $K_{s}(R)$. In [7], it is proved that $K_{s}(R)$ is prime if and only if $R$ is prime and $s\neq 0$, and $K_{s}(R)$ is semiprime if and only if $R$ is semiprime and $s$ is a non zero-divisor. Also prime ideals of $K_{s}(R)$ are determined. Motivated by these, in this paper, we study on the primeness and semiprimeness of a Morita context in a general setting. We determine equivalent conditions for an ideal to be a prime ideal or a semiprime ideal in a Morita context. In particular, prime radical of a Morita context is characterized. We obtain necessary and sufficient conditions for a Morita context to be a prime ring and a semiprime ring.

In what follows, $\mathbb{Z}$ and $\mathbb{Z}_{n}$ denote the ring of integers and the ring of integers modulo $n$ for some positive integer $n$, respectively. Also \left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right) denotes the subset of a Morita context \left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right) consisting of all matrices \left(\begin{array}[]{cc}r&v\\ w&s\\ \end{array}\right) with $r\in I$, $s\in J$, $v\in V_{1}$, $w\in W_{1}$ where $I,V_{1},W_{1},J$ are subsets of $R,V,W,S$, respectively.

2. Primeness of a Morita Context

In this section, we work on ideal structures in a Morita context in terms of primeness and semiprimeness. We exhibit some results in order to prove our main theorems. In Lemma 2.1 for a Morita context $(R,V,W,S)$, elements of $R\oplus W$ and of $V\oplus S$ are written as column vectors, and elements of $R\oplus V$ and of $W\oplus S$ are written as row vectors. For unexplained notations and definitions we refer to [7]. The following result is very useful for this paper.

Lemma 2.1**.**

[7, Lemma 2.1]** Let $(R,V,W,S)$ be a Morita context and $\emptyset\neq U\subseteq(R,V,W,S)$. Then the following hold.

1. (1)

$U$* is a right ideal of $(R,V,W,S)$ if and only if $U=(C_{1}(U)C_{2}(U))$ where $C_{1}(U)$ is a right $R$-submodule of $R\oplus W$ and $C_{2}(U)$ is a right $S$-submodule of $V\oplus S$ with $C_{1}(U)V\subseteq C_{2}(U)$ and $C_{2}(U)W\subseteq C_{1}(U)$.* 2. (2)

$U$* is a left ideal of $(R,V,W,S)$ if and only if U=\left(\begin{array}[]{c}R_{1}(U)\\ R_{2}(U)\\ \end{array}\right) where $R_{1}(U)$ is a left $R$-submodule of $R\oplus V$ and $R_{2}(U)$ is left $S$-submodule of $W\oplus S$ with $VR_{2}(U)\subseteq R_{1}(U)$ and $WR_{1}(U)\subseteq R_{2}(U)$.* 3. (3)

$U$* is an ideal of $(R,V,W,S)$ if and only if $U=(I,V_{1},W_{1},J)$ where $I\trianglelefteq R$, $J\trianglelefteq S$, $V_{1}\leqslant~{}_{R}V_{S}$ and $W_{1}\leqslant~{}_{S}W_{R}$ with $V_{1}W\subseteq I$, $W_{1}V\subseteq J$, $IV\subseteq V_{1}$, $JW\subseteq W_{1}$, $VW_{1}\subseteq I$, $WV_{1}\subseteq J$, $VJ\subseteq V_{1}$, $WI\subseteq W_{1}$.*

Lemma 2.2**.**

Let $T=(R,V,W,S)$ be a Morita context and $U$ a right ideal of $T$. Then the following hold.

1. (1)

If \left(\begin{array}[]{c}r\\ w\\ \end{array}\right)\in C_{1}(U), then \left(\begin{array}[]{cc}r&0\\ w&0\\ \end{array}\right)\in U, 2. (2)

If \left(\begin{array}[]{c}v\\ s\\ \end{array}\right)\in C_{2}(U), then \left(\begin{array}[]{cc}0&v\\ 0&s\\ \end{array}\right)\in U.

Proof.

$(1)$ Let \left(\begin{array}[]{c}r\\ w\\ \end{array}\right)\in C_{1}(U). By [7], there exists \left(\begin{array}[]{cc}r&v\\ w&s\\ \end{array}\right)\in U. For \left(\begin{array}[]{cc}1&0\\ 0&0\\ \end{array}\right)\in T, as $U$ is a right ideal of $T$, we have \left(\begin{array}[]{cc}r&v\\ w&s\\ \end{array}\right)\left(\begin{array}[]{cc}1&0\\ 0&0\\ \end{array}\right)=\left(\begin{array}[]{cc}r&0\\ w&0\\ \end{array}\right)\in U. The proof of (2) is similar to that of (1). ∎

We now investigate primeness for a one sided ideal of a Morita context.

Proposition 2.3**.**

Let $(R,V,W,S)$ be a Morita context and $U$ a right ideal of $(R,V,W,S)$. If $U$ is a prime right ideal of $(R,V,W,S)$, then $C_{1}(U)$ and $C_{2}(U)$ are prime submodules of right $R$-module $R\oplus W$ and right $S$-module $V\oplus S$, respectively.

Proof.

In order to prove $C_{1}(U)$ is a prime submodule of $(R\oplus W)_{R}$, let $r\in R$ and \left(\begin{array}[]{c}m_{1}\\ m_{2}\\ \end{array}\right)\in R\oplus W with \left(\begin{array}[]{c}m_{1}\\ m_{2}\\ \end{array}\right)Rr\subseteq C_{1}(U). We show that $(R\oplus W)r\subseteq C_{1}(U)$ or \left(\begin{array}[]{c}m_{1}\\ m_{2}\\ \end{array}\right)\in C_{1}(U). Let \left(\begin{array}[]{c}a\\ w\\ \end{array}\right)\in R\oplus W. Then for any \left(\begin{array}[]{cc}x&y\\ z&t\\ \end{array}\right)\in(R,V,W,S), being \left(\begin{array}[]{c}m_{1}\\ m_{2}\\ \end{array}\right)(xa+yw)r\in C_{1}(U) and Lemma 2.2 imply that \left(\begin{array}[]{cc}m_{1}&0\\ m_{2}&0\\ \end{array}\right)\left(\begin{array}[]{cc}x&y\\ z&t\\ \end{array}\right)\left(\begin{array}[]{cc}ar&0\\ wr&0\\ \end{array}\right)=\left(\begin{array}[]{cc}m_{1}(xa+yw)r&0\\ m_{2}(xa+yw)r&0\\ \end{array}\right)\in U. Since $U$ is a prime right ideal of $(R,V,W,S)$, \left(\begin{array}[]{cc}m_{1}&0\\ m_{2}&0\\ \end{array}\right)\in U or \left(\begin{array}[]{cc}ar&0\\ wr&0\\ \end{array}\right)\in U. Hence \left(\begin{array}[]{c}m_{1}\\ m_{2}\\ \end{array}\right)\in C_{1}(U) or \left(\begin{array}[]{c}a\\ w\\ \end{array}\right)r\in C_{1}(U), as asserted. Similarly, it can be shown that $C_{2}(U)$ is a prime right $S$-submodule of $V\oplus S$. ∎

Proposition 2.3 is useful to determine a right ideal in a Morita context is prime or not.

Example 2.4**.**

Consider the Morita context $T=(\mathbb{Z}_{8},\mathbb{Z}_{8},\mathbb{Z}_{8},\mathbb{Z}_{8})$, where the context products are the same as the product in $\mathbb{Z}_{8}$. Take $U=(\overline{4}\mathbb{Z}_{8},\overline{4}\mathbb{Z}_{8},\mathbb{Z}_{8},\mathbb{Z}_{8})$ as a right ideal of $T$. Note that $\overline{4}\mathbb{Z}_{8}\oplus\mathbb{Z}_{8}=C_{1}(U)$. For $\overline{2}\in\mathbb{Z}_{8}$ and \left(\begin{array}[]{c}\overline{2}\\ \overline{2}\\ \end{array}\right)\in\mathbb{Z}_{8}\oplus\mathbb{Z}_{8}, we have \overline{2}\mathbb{Z}_{8}\left(\begin{array}[]{c}\overline{2}\\ \overline{2}\\ \end{array}\right)\subseteq C_{1}(U). Also, \left(\begin{array}[]{c}\overline{2}\\ \overline{2}\\ \end{array}\right)\notin C_{1}(U) and $\overline{2}(\mathbb{Z}_{8}\oplus\mathbb{Z}_{8})\nsubseteq C_{1}(U)$. Hence $C_{1}(U)$ is not a prime submodule of $R\oplus W$. Therefore $U$ is not a right prime ideal of $T$.

Lemma 2.5**.**

Let $(R,V,W,S)$ be a Morita context. If $(I,V_{1},W_{1},J)$ is a semiprime ideal of $(R,V,W,S)$, then we have

1. (1)

$\{v\in V~{}:~{}vW\subseteq I\}=\{v\in V~{}:~{}Wv\subseteq J\}$, 2. (2)

$\{w\in W~{}:~{}Vw\subseteq I\}=\{w\in W~{}:~{}wV\subseteq J\}$.

Proof.

(1) Let $A=\{v\in V~{}:~{}vW\subseteq I\}$ and $B=\{v\in V~{}:~{}Wv\subseteq J\}$. Assume $v\in A$. Then $vW\subseteq I$ and so for any $w\in W$ the product \left(\begin{array}[]{cc}vw&0\\ 0&wv\\ \end{array}\right)\left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right)\left(\begin{array}[]{cc}vw&0\\ 0&wv\\ \end{array}\right)=\left(\begin{array}[]{cc}vwRvw&vwVwv\\ wvWvw&wvSwv\\ \end{array}\right) is contained by \left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). Since \left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right) is a semiprime ideal of the Morita context, \left(\begin{array}[]{cc}vw&0\\ 0&wv\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). Hence $Wv\subseteq J$ and so $v\in B$. Thus $A\subseteq B$. Similarly, it can be shown that $B\subseteq A$. $(2)$ can be proved via similar techniques. ∎

Proposition 2.6**.**

Let $(R,V,W,S)$ be a Morita context, $I$ and $J$ are prime ideals of $R$ and $S$, respectively. Then the following hold.

1. (1)

$\{v\in V:vW\subseteq I\}$* is a prime left $R$-submodule of $V$.* 2. (2)

$\{v\in V:Wv\subseteq J\}$* is a prime right $S$-submodule of $V$.* 3. (3)

$\{w\in W:Vw\subseteq I\}$* is a prime right $R$-submodule of $W$.* 4. (4)

$\{w\in W:wV\subseteq J\}$* is a prime left $S$-submodule of $W$.*

Proof.

(1) Let $A$ denote the left $R$-submodule $\{v\in V:vW\subseteq I\}$ of $V$, and $r\in R$, $v\in V$ with $rRv\subseteq A$. Then $rRvW\subseteq I$. Since $I$ is a prime ideal of $I$, $r\in I$ or $vW\subseteq I$. If $r\in I$, then $rVW\subseteq rR\subseteq I$. Hence $rV\subseteq A$ or $v\in A$. Therefore $A$ is prime.

(2) Let $B$ denote the right $S$-submodule $\{v\in V:Wv\subseteq J\}$ of $V$, and $s\in S$, $v\in V$ with $vSs\subseteq B$. Then $WvSs\subseteq J$. The ideal $J$ being prime implies that $s\in J$ or $Wv\subseteq J$. If $s\in J$, then $WVs\subseteq Ss\subseteq J$. Thus $Vs\subseteq B$ or $v\in B$. Therefore $B$ is prime.

(3) and (4) are proved similarly. ∎

We now give the following result related to an ideal of a Morita context to be prime.

Theorem 2.7**.**

Let $T=(R,V,W,S)$ be a Morita context and consider the following conditions for an ideal $(I,V_{1},W_{1},J)$ of $T$.

1. (1)

$(I,V_{1},W_{1},J)$* is prime.* 2. (2)

$I$, $J$ are prime ideals of $R$ and $S$, respectively, and $V_{1}=\{v\in V:vW\subseteq I\}=\{v\in V:Wv\subseteq J\}$ and $W_{1}=\{w\in W:Vw\subseteq I\}=\{w\in W:wV\subseteq J\}$.

Then (1) $\Rightarrow$ (2). If $VW=R$ and $WV=S$, then (2) $\Rightarrow$ (1).

Proof.

(1) $\Rightarrow$ (2) Assume that $(I,V_{1},W_{1},J)$ is a prime ideal of $T$. In order to see $I$ is a prime ideal of $R$, let $r_{1},r_{2}\in R$ with $r_{1}Rr_{2}\subseteq I$. We have \left(\begin{array}[]{cc}r_{1}&0\\ 0&0\\ \end{array}\right)\left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right)\left(\begin{array}[]{cc}r_{2}&0\\ 0&0\\ \end{array}\right)=\left(\begin{array}[]{cc}r_{1}Rr_{2}&0\\ 0&0\\ \end{array}\right). So \left(\begin{array}[]{cc}r_{1}Rr_{2}&0\\ 0&0\\ \end{array}\right)\subseteq\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). Since $(I,V_{1},W_{1},J)$ is a prime ideal, \left(\begin{array}[]{cc}r_{1}&0\\ 0&0\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right) or \left(\begin{array}[]{cc}r_{2}&0\\ 0&0\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). Then $r_{1}\in I$ or $r_{2}\in I$. Hence $I$ is a prime ideal of $R$. Similarly, it can be shown that $J$ is a prime ideal of $S$. Let $A=\{v\in V~{}:~{}vW\subseteq I\}$. By Lemma 2.1, it is known that $V_{1}\subseteq A=\{v\in V~{}:~{}vW\subseteq I\}$. It is enough to show that $A\subseteq V_{1}$. If $x\in A$, then $xW\subseteq I$ and so

[TABLE]

Since $(I,V_{1},W_{1},J)$ is prime, \left(\begin{array}[]{cc}0&x\\ 0&0\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). So $x\in V_{1}$. By Lemma 2.5, $V_{1}=\{v\in V~{}:~{}vW\subseteq I\}=\{v\in V~{}:~{}Wv\subseteq J\}$. Similarly, one can show that $W_{1}=\{w\in W~{}:~{}Vw\subseteq I\}=\{w\in W~{}:~{}wV\subseteq J\}$.

(2) $\Rightarrow$ (1) Let \left(\begin{array}[]{cc}r_{1}&v_{1}\\ w_{1}&s_{1}\\ \end{array}\right),\left(\begin{array}[]{cc}r_{2}&v_{2}\\ w_{2}&s_{2}\\ \end{array}\right)\in T such that

[TABLE]

Then

[TABLE]

is contained by \left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). By $(1,1)$ entry of above matrix, $r_{1}Rr_{2}\subseteq I$. As $I$ is prime, $r_{1}\in I$ or $r_{2}\in I$. Hence we have the following cases.

Case I: Suppose that $r_{1},r_{2}\in I$. By $(1,1)$ entry of above matrix, $v_{1}Sw_{2}\subseteq I$, and so $v_{1}WVw_{2}\subseteq I$. Being $v_{1}WRVw_{2}R\subseteq I$ implies $v_{1}WR\subseteq I$ or $Vw_{2}R\subseteq I$. Thus $v_{1}\in V_{1}$ or $w_{2}\in W_{1}$. Therefore we have the following subcases:

1. I(a)

Suppose $v_{1}\notin V_{1}$ and $w_{2}\in W_{1}$. By $(1,2)$ entry of above matrix, $v_{1}Ss_{2}\subseteq V_{1}$. By Proposition 2.6, the right $S$-submodule $V_{1}$ being prime implies that $v_{1}\in V_{1}$ or $Vs_{2}\subseteq V_{1}$. By supposition, we have only $Vs_{2}\subseteq V_{1}$. Then $WVs_{2}=Ss_{2}\subseteq WV_{1}\subseteq J$, so $s_{2}\in J$. On the other hand, again by $(1,2)$ entry of above matrix, $v_{1}Wv_{2}\subseteq V_{1}$. Since $v_{1}Wv_{2}W\subseteq V_{1}W\subseteq I$ and $v_{1}\notin V_{1}$, $v_{2}W\subseteq I$ due to the primeness of $I$. Hence $v_{2}\in V_{1}$. Therefore \left(\begin{array}[]{cc}r_{2}&v_{2}\\ w_{2}&s_{2}\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). 2. I(b)

Suppose $v_{1}\in V_{1}$ and $w_{2}\notin W_{1}$. By $(2,1)$ entry of above matrix, $s_{1}Sw_{2}\subseteq W_{1}$. The left $S$-submodule $W_{1}$ being prime and $w_{2}\notin W_{1}$ imply that $s_{1}W\subseteq W_{1}$. So $s_{1}WV=s_{1}S\subseteq W_{1}V\subseteq J$. Hence $s_{1}\in J$. On the other hand, again by $(2,1)$ entry of above matrix, $w_{1}Vw_{2}\subseteq W_{1}$. Then $w_{1}Vw_{2}V\subseteq W_{1}V\subseteq J$ implies $w_{1}V\subseteq J$ or $w_{2}V\subseteq J$, and so $w_{1}\in W_{1}$ or $w_{2}\in W_{1}$. By supposition, the only possibility is $w_{1}\in W_{1}$. Therefore \left(\begin{array}[]{cc}r_{1}&v_{1}\\ w_{1}&s_{1}\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). 3. I(c)

Suppose $v_{1}\in V_{1}$ and $w_{2}\in W_{1}$. By $(2,1)$ entry of above matrix, $w_{1}Rr_{2}\subseteq W_{1}$. Since $W_{1}$ is a prime right $R$-submodule of $W$, $w_{1}\in W_{1}$ or $Wr_{2}\subseteq W_{1}$. Being $Wr_{2}\subseteq W_{1}$ implies that $VWr_{2}\subseteq VW_{1}\subseteq I$, and so $r_{2}\in I$, but we have this fact at the beginning of Case I. It follows that $w_{1}\in W_{1}$ or $w_{1}\notin W_{1}$.

1. I(c)-i

Suppose $w_{1}\in W_{1}$. By $(2,2)$ entry of above matrix, $s_{1}Wv_{2}\subseteq J$. Then $Ss_{1}SWv_{2}\subseteq J$. The primeness of $J$ implies that $s_{1}\in J$ or $Wv_{2}\subseteq J$, so $v_{2}\in V_{1}$.

• •

If $s_{1}\in J$, then \left(\begin{array}[]{cc}r_{1}&v_{1}\\ w_{1}&s_{1}\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right).

• •

If $s_{1}\notin J$, then $v_{2}\in V_{1}$. By $(2,2)$ entry of above matrix, $s_{1}Ss_{2}\subseteq J$. By supposition and the primeness of $J$, we have $s_{2}\in J$. Thus \left(\begin{array}[]{cc}r_{2}&v_{2}\\ w_{2}&s_{2}\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). 2. I(c)-ii

Suppose that $w_{1}\notin W_{1}$. By $(2,2)$ entry of above matrix, $w_{1}Rv_{2}=w_{1}VWv_{2}\subseteq J$. Since $J$ is prime and $w_{1}VSWv_{2}S\subseteq J$, $w_{1}V\subseteq J$ or $Wv_{2}\subseteq J$. It follows that $w_{1}\in W_{1}$ or $v_{2}\in V_{1}$. By supposition, we only have $v_{2}\in V_{1}$. On the other hand, again by $(2,2)$ entry of above matrix, $w_{1}Vs_{2}\subseteq J$. Then $w_{1}VSs_{2}S\subseteq J$. The primeness of $J$ implies $w_{1}V\subseteq J$ or $s_{2}\in J$. Since $w_{1}\notin W_{1}$, we have $s_{2}\in J$. Therefore \left(\begin{array}[]{cc}r_{2}&v_{2}\\ w_{2}&s_{2}\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right).

Case II: Without loss of generality, suppose that $r_{1}\notin I$ and $r_{2}\in I$. By $(1,1)$ entry of above matrix, $r_{1}Vw_{2}\subseteq I$. Hence $r_{1}RVw_{2}R\subseteq I$. Since $I$ is a prime ideal, $r_{1}R\subseteq I$ or $Vw_{2}\subseteq I$. As $r_{1}\notin I$, $Vw_{2}\subseteq I$. So $w_{2}\in W_{1}$. By $(1,2)$ entry of above matrix, $r_{1}Rv_{2}\subseteq V_{1}$. Since $V_{1}$ is a prime right $R$-submodule of $V$, $r_{1}V\subseteq V_{1}$ or $v_{2}\in V_{1}$. Assume that $r_{1}V\subseteq V_{1}$. Hence $r_{1}VW=r_{1}R\subseteq I$. This contradiction implies that $v_{2}\in V_{1}$. Lastly, from $(1,2)$ entry of above matrix, $r_{1}Vs_{2}\subseteq V_{1}$. Thus $r_{1}Vs_{2}W\subseteq I$. Then $r_{1}RVs_{2}W\subseteq I$. This implies that $r_{1}R\subseteq I$ or $Vs_{2}W\subseteq I$, as I is prime. Since $r_{1}R\nsubseteq I$, $Vs_{2}W\subseteq I$. Thus $Vs_{2}WV=Vs_{2}S\subseteq IV\subseteq V_{1}$. So $WVs_{2}S=Ss_{2}S\subseteq WV_{1}\subseteq J$. Then $s_{2}\in J$ and therefore \left(\begin{array}[]{cc}r_{2}&v_{2}\\ w_{2}&s_{2}\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). This completes the proof. ∎

The conditions $VW=R$ and $WV=S$ are not superfluous in Theorem 2.7 ($(2)\Rightarrow(1)$).

Example 2.8**.**

Consider the Morita context T=\left(\begin{array}[]{cc}\mathbb{Z}_{6}&\overline{2}\mathbb{Z}_{6}\\ \overline{3}\mathbb{Z}_{6}&\mathbb{Z}_{6}\\ \end{array}\right), where the context products are the same as the product in $\mathbb{Z}_{6}$. Then $VW=0$ and $WV=0$. For the ideal H=\left(\begin{array}[]{cc}\overline{3}\mathbb{Z}_{6}&\overline{2}\mathbb{Z}_{6}\\ \overline{3}\mathbb{Z}_{6}&\overline{2}\mathbb{Z}_{6}\\ \end{array}\right) of $T$, the entries of $H$ satisfy all conditions in Theorem 2.7 (2). Although

[TABLE]

\left(\begin{array}[]{cc}\overline{3}&\overline{0}\\ \overline{0}&\overline{3}\\ \end{array}\right)\notin H* and \left(\begin{array}[]{cc}\overline{1}&\overline{0}\\ \overline{0}&\overline{2}\\ \end{array}\right)\notin H. Therefore $H$ is not a prime ideal of $T$.*

The following theorem is proved in [6, Theorem 1] by Sands. We give a short proof by using prime ideals determined in Theorem 2.7.

Theorem 2.9**.**

Let $T=(R,V,W,S)$ be a Morita context and $P(T)$ the prime radical of $T$. Then P(T)=\left(\begin{array}[]{cc}P(R)&V_{0}\\ W_{0}&P(S)\\ \end{array}\right), where $V_{0}=\{v\in V~{}:~{}vW\subseteq P(R)\}=\{v\in V~{}:~{}Wv\subseteq P(S)\}$ and $W_{0}=\{w\in W~{}:~{}wV\subseteq P(S)\}=\{w\in W~{}:~{}Vw\subseteq P(R)\}$.

Proof.

Let $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$ be prime spectrums of $R$ and $S$, respectively. Let $I\in\mathcal{K}_{1}$ and $J\in\mathcal{K}_{2}$. Also, $V_{I}=\{v\in V~{}:~{}vW\subseteq I\}=\{v\in V~{}:~{}Wv\subseteq J\}$ and $W_{J}=\{w\in W~{}:~{}wV\subseteq J\}=\{w\in W~{}:~{}Vw\subseteq I\}$. Then $(I,V_{I},W_{J},J)$ is a prime ideal of $T$. Let $U$ be a prime ideal of $T$. Then $U$ has the form $(I,V_{I},W_{J},J)$ for some $I\in\mathcal{K}_{1}$, $J\in\mathcal{K}_{2}$ by Lemma 2.1, Lemma 2.5 and Theorem 2.7. If all of the ideals $(I,V_{I},W_{J},J)$ for $I\in\mathcal{K}_{1}$, $J\in\mathcal{K}_{2}$ are intersected, then the result is obtained by an easy calculation. ∎

Note that $V/V_{0}$ is a left $R/P(R)$ right $S/P(S)$-bimodule and $W/W_{0}$ is a left $S/P(S)$ right $R/P(R)$-bimodule. Hence $(R/P(R),V/V_{0},W/W_{0},S/P(S))$ is a Morita context where the context products are given by

[TABLE]

for all $v\in V$ and $w\in W$. Then we have the next result.

Proposition 2.10**.**

Let $T=(R,V,W,S)$ be a Morita context, and \left(\begin{array}[]{cc}R/P(R)&V/V_{0}\\ W/W_{0}&S/P(S)\\ \end{array}\right) be defined above. Then

[TABLE]

Proof.

Let $f$ denote the map from $T/P(T)$ to \left(\begin{array}[]{cc}R/P(R)&V/V_{0}\\ W/W_{0}&S/P(S)\\ \end{array}\right) defined by \left(\begin{array}[]{cc}r&v\\ w&s\\ \end{array}\right)+P(T)\mapsto\left(\begin{array}[]{cc}r+P(R)&v+V_{0}\\ w+W_{0}&s+P(S)\\ \end{array}\right). By easy calculations, it can be shown that $f$ is a ring isomorphism. ∎

Now we give a result related to an ideal of a Morita context to be a semiprime ideal.

Theorem 2.11**.**

Let $T=(R,V,W,S)$ be a Morita context. Then the following are equivalent for an ideal $(I,V_{1},W_{1},J)$ of $T$.

1. (1)

$(I,V_{1},W_{1},J)$* is semiprime.* 2. (2)

$I$* and $J$ are semiprime ideals of $R$ and $S$, respectively, and $V_{1}=\{v\in V:vW\subseteq I\}=\{v\in V:Wv\subseteq J\}$ and $W_{1}=\{w\in W:Vw\subseteq I\}=\{w\in W:wV\subseteq J\}$.*

Proof.

(1) $\Rightarrow$ (2) In order to see $I$ is a semiprime ideal of $R$, let $r\in R$ such that $rRr\subseteq I$. Then we have \left(\begin{array}[]{cc}r&0\\ 0&0\\ \end{array}\right)\left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right)\left(\begin{array}[]{cc}r&0\\ 0&0\\ \end{array}\right)=\left(\begin{array}[]{cc}rRr&0\\ 0&0\\ \end{array}\right) \subseteq\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). Since \left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right) is a semiprime ideal of $T$, \left(\begin{array}[]{cc}r&0\\ 0&0\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). Hence $r\in I$, and so $I$ is a semiprime ideal of $R$. Similarly, one can show that $J$ is a semiprime ideal of $S$. By Lemma 2.1, it is known that $V_{1}\subseteq A=\{v\in V~{}:~{}vW\subseteq I\}$. It is enough to show $A\subseteq V_{1}$. Take $x\in A$. So $xW\subseteq I$. Hence \left(\begin{array}[]{cc}0&x\\ 0&0\\ \end{array}\right)\left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right)\left(\begin{array}[]{cc}0&x\\ 0&0\\ \end{array}\right)=\left(\begin{array}[]{cc}0&xWx\\ 0&0\\ \end{array}\right)\subseteq\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). The ideal \left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right) being semiprime implies that \left(\begin{array}[]{cc}0&x\\ 0&0\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right) and so $x\in V_{1}$. Thus $A\subseteq V_{1}$. By Lemma 2.5, $V_{1}=\{v\in V~{}:~{}vW\subseteq I\}=\{v\in V~{}:~{}Wv\subseteq J\}$. Similarly, one can show that $W_{1}=\{w\in W~{}:~{}Vw\subseteq I\}=\{w\in W~{}:~{}wV\subseteq J\}$.

(2) $\Rightarrow$ (1) Let \left(\begin{array}[]{cc}r&v\\ w&s\\ \end{array}\right)\in T with \left(\begin{array}[]{cc}r&v\\ w&s\\ \end{array}\right)\left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right)\left(\begin{array}[]{cc}r&v\\ w&s\\ \end{array}\right)=

\left(\begin{array}[]{cc}rRr+vWr+rVw+vSw&rRv+vWv+rVs+vSs\\ wRr+sWr+wVw+sSw&wRv+sWv+wVs+sSs\\ \end{array}\right) \subseteq\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). By $(1,1)$ entry of above matrix, $rRr\subseteq I$. As $I$ is semiprime, $r\in I$. Similarly $s\in J$. By $(2,1)$ entry of above matrix $wVw\subseteq W_{1}$. Then $wVwV=(wV)^{2}\subseteq W_{1}V\subseteq J$. Since $J$ is semiprime, $wV\subseteq J$. So $w\in W_{1}$. Also by $(1,2)$ entry of above matrix, $vWv\subseteq V_{1}$. Then $vWvW=(vW)^{2}\subseteq V_{1}W\subseteq I$. Since $I$ is a semiprime ideal of $R$, $vW\subseteq I$. Thus $v\in V_{1}$. Therefore \left(\begin{array}[]{cc}r&v\\ w&s\\ \end{array}\right)\in\left(\begin{array}[]{cc}I&V_{1}\\ W_{1}&J\\ \end{array}\right). This completes the proof. ∎

In Theorem 2.11(2), semiprimeness of the ideals $I$ and $J$ in the rings $R$ and $S$, respectively, are not superfluous as the following example shows.

Example 2.12**.**

Consider T=\left(\begin{array}[]{cc}\mathbb{Z}_{4}&\overline{2}\mathbb{Z}_{4}\\ \overline{2}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ \end{array}\right) as a Morita context, where the context products are the same as the product in $\mathbb{Z}_{4}$. It is clear that H=\left(\begin{array}[]{cc}\overline{2}\mathbb{Z}_{4}&\overline{2}\mathbb{Z}_{4}\\ \overline{2}\mathbb{Z}_{4}&\{\overline{0}\}\\ \end{array}\right) is an ideal of $T$. Also \left(\begin{array}[]{cc}\overline{0}&\overline{0}\\ \overline{0}&\overline{2}\\ \end{array}\right)\left(\begin{array}[]{cc}\mathbb{Z}_{4}&\overline{2}\mathbb{Z}_{4}\\ \overline{2}\mathbb{Z}_{4}&\mathbb{Z}_{4}\\ \end{array}\right)\left(\begin{array}[]{cc}\overline{0}&\overline{0}\\ \overline{0}&\overline{2}\\ \end{array}\right)=\left(\begin{array}[]{cc}\overline{0}&\overline{0}\\ \overline{0}&\overline{0}\\ \end{array}\right)\in H. But \left(\begin{array}[]{cc}\overline{0}&\overline{0}\\ \overline{0}&\overline{2}\\ \end{array}\right)\notin H. Therefore $H$ is not semiprime ideal of $T$, also $\{\overline{0}\}$ is not semiprime in $\mathbb{Z}_{4}$.

In the sequel, we determine under what conditions a Morita context is prime or semiprime.

Theorem 2.13**.**

Let $(R,V,W,S)$ be a Morita context and consider the following conditions.

1. (1)

$(R,V,W,S)$* is a prime ring.* 2. (2)

$R$, $S$ are prime rings and ${}_{R}V_{S}$, ${}_{S}W_{R}$ are prime modules. 3. (3)

$R$* and $S$ are prime rings.* 4. (4)

$R$* or $S$ is a prime ring.*

Then (1) $\Rightarrow$ (2) $\Rightarrow$ (3)$\Rightarrow$ (4). If $VW=R$ and $WV=S$, then (4) $\Rightarrow$ (1).

Proof.

(1) $\Rightarrow$ (2) Let $r_{1},r_{2}\in R$ such that $r_{1}Rr_{2}=0$. Then we have \left(\begin{array}[]{cc}r_{1}&0\\ 0&0\\ \end{array}\right)\left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right)\left(\begin{array}[]{cc}r_{2}&0\\ 0&0\\ \end{array}\right)= \left(\begin{array}[]{cc}r_{1}Rr_{2}&0\\ 0&0\\ \end{array}\right). As $(R,V,W,S)$ is prime, $r_{1}=0$ or $r_{2}=0$. Hence $R$ is a prime ring. Similarly, it can be shown that $S$ is prime. Now let $r\in R$ and $v\in V$ with $rRv=0$. Then \left(\begin{array}[]{cc}r&0\\ 0&0\\ \end{array}\right)\left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right)\left(\begin{array}[]{cc}0&v\\ 0&0\\ \end{array}\right)= \left(\begin{array}[]{cc}0&rRv\\ 0&0\\ \end{array}\right). Thus $r=0$ or $v=0$. This implies that $rV=0$ or $v=0$. Hence $V$ is prime as a left $R$-module. On the other hand, let $s\in S$ and $v\in V$ with $vSs=0$. Since \left(\begin{array}[]{cc}0&v\\ 0&0\\ \end{array}\right)\left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right)\left(\begin{array}[]{cc}0&0\\ 0&s\\ \end{array}\right)= \left(\begin{array}[]{cc}0&vSs\\ 0&0\\ \end{array}\right), by (1), $v=0$ or $s=0$. Thus $v=0$ or $Vs=0$. It follows that $V$ is also prime as a right $S$-module. By the same technique, it can be shown that ${}_{S}W_{R}$ is a prime module.

(2) $\Rightarrow$ (3) and (3) $\Rightarrow$ (4) are obvious.

(4) $\Rightarrow$ (1) Assume that $R$ is prime. Let $VsW=0$ for $s\in S$. By hypothesis, $WVsWV=SsS=0$. So $s=0$. By [1, Theorem 27], we have $S$ is prime. If $S$ is prime, similar proof can work for $R$. We claim that 0=\left(\begin{array}[]{cc}0_{R}&0_{V}\\ 0_{W}&0_{S}\\ \end{array}\right) is a prime ideal of \left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right). Since $R$ and $S$ are prime rings, $0_{R}$ and $0_{S}$ are prime ideals of $R$ and $S$, respectively. Now consider the sets $\{v\in V:vW=0\}$ and $\{v\in V:Wv=0\}$. Let $v\in\{v\in V:vW=0\}$. Then $vW=0$. By hypothesis, $vWV=vS=0$, and so $v=0$. Hence $\{v\in V:vW=0\}=0$. On the other hand, being $VW=R$ implies $\{v\in V:Wv=0\}=0$. Similarly, $\{w\in W:Vw=0\}=\{w\in W:wV=0\}=0$. Therefore [math] is a prime ideal of \left(\begin{array}[]{cc}R&V\\ W&S\\ \end{array}\right) by Theorem 2.7. This completes the proof. ∎

Theorem 2.14**.**

Let $T=(R,V,W,S)$ be a Morita context and consider the following conditions.

1. (1)

$(R,V,W,S)$* is a semiprime ring.* 2. (2)

$R$* and $S$ are semiprime rings.* 3. (3)

$R$* or $S$ is a semiprime ring.*

Then (1) $\Rightarrow$ (2)$\Rightarrow$ (3). If $VW=R$ and $WV=S$, then (3) $\Rightarrow$ (1).

Proof.

(1) $\Rightarrow$ (2) It is similar to proof of Theorem 2.13 (1) $\Rightarrow$ (2).

(2) $\Rightarrow$ (3) It is clear.

(3) $\Rightarrow$ (1) Assume that $R$ is a semiprime ring. Let $VsW=0$ for $s\in S$. Via similar discussion to proof of Theorem 2.13, we have $s=0$. So $S$ is semiprime, by [1, Corollary 21]. Similarly we can prove that $R$ is semiprime, if $S$ is semiprime. If we show that $P(T)=0$, then the proof is completed. By Theorem 2.9, P(T)=\left(\begin{array}[]{cc}P(R)&V_{0}\\ W_{0}&P(S)\\ \end{array}\right) where $V_{0}=\{v\in V:vW\subseteq P(R)\}=\{v\in V:Wv\subseteq P(S)\}$ and $W_{0}=\{w\in W:wV\subseteq P(S)\}=\{w\in W:Vw\subseteq P(R)\}$. As $R$ and $S$ are semiprime, $P(R)=0$ and $P(S)=0$. We claim that $V_{0}=0$ and $W_{0}=0$. Let $v\in V_{0}$. Then $vW=0$, and so $vWV=vS=0$. Hence $v=0$, and thus $V_{0}=0$. Similarly, one can show that $W_{0}=0$. Therefore $P(T)=0$, as desired. ∎