Some bounds for commuting probability of finite rings

TL;DR

This paper investigates the likelihood that two randomly selected elements of a finite ring commute, providing bounds on this probability to deepen understanding of ring structure.

Contribution

It introduces new bounds for the commuting probability of finite rings, enhancing theoretical understanding of their algebraic properties.

Findings

Derived upper and lower bounds for commuting probability

Provided insights into the structure of finite rings based on commuting probability

Extended previous results with tighter bounds

Abstract

Let $R$ be a finite ring. The commuting probability of $R$ is the probability that any two randomly chosen elements of $R$ commute. In this paper, we obtain some bounds for commuting probability of $R$.

Equations

$$\mathrm{Pr}(R)=\frac{|\{(s,r)\in R\times R:sr=rs\}|}{|R\times R|}.$$

$$\mathrm{Pr}(R)=\frac{1}{|R{|}^2}\sum _{r\in R}|C_R(r)|$$

$$\mathrm{Pr}(R)=\frac{|Z(R)|}{|R|}+\frac{1}{|R{|}^2}\sum _{r\in R\setminus Z(R)}|C_R(r)|.$$

$$\mathrm{Pr}(R)\le \frac{p^2+p-1}{p^3}.$$

$$|R{|}^2\mathrm{Pr}(R)=|R||Z(R)|+\sum _{r\in R\setminus Z(R)}|C_R(r)|.$$

$$\sum _{r\in R\setminus Z(R)}|C_R(r)|\ge p(|R|-|Z(R)|)$$

$$\sum _{r\in R\setminus Z(R)}|C_R(r)|\le \frac{|R|(|R|-|Z(R)|)}{p}$$

$$\frac{(p-1)|Z(R)|+|R|}{p|R|}\le \frac{p^2+p-1}{p^3}.$$

$$\mathrm{Pr}(R)\le \mathrm{Pr}(S).$$

$$\frac{C_R(x)+N}{N}\subseteq C_{R/N}(x+N)\text{for all}x\in R.$$

$$(s+N)(x+N)=rx+N=xr+N=(x+N)(s+N),$$

$$\mathrm{Pr}(R)\le \mathrm{Pr}(R/N)\mathrm{Pr}(N).$$

$$|R{|}^2\mathrm{Pr}(R)=$$

$$=$$

$$\le$$

$$=$$

$$|R{|}^2\mathrm{Pr}(R)\le$$

$$=$$

$$\frac{C_R(x)+N}{N}=C_{R/N}(x+N)\text{for all}x\in R.$$

$$\mathrm{Pr}(R)>\frac{1}{|[R,R]|}.$$

$$\mathrm{Pr}(R)\ge \frac{1}{|K(S,R)|}\left(1+\frac{|K(R,R)|-1}{|R:Z(R)|}\right)$$

$$\mathrm{Pr}(R)=\frac{|Z(R)|}{|R|}+\frac{1}{|R|}\sum _{r\in R\setminus Z(R)}\frac{1}{|R:C_R(r)|}.$$

$$\mathrm{Pr}(R)\ge$$

$$=$$

$$\mathrm{Pr}(R)\ge \frac{1}{|[R,R]|}\left(1+\frac{|[R,R]|-1}{|R:Z(R)|}\right)$$

$$\mathrm{Pr}(R)=\frac{|Z(R)|}{|R|}+\frac{1}{|R|}\sum _{r\in R\setminus Z(R)}\frac{1}{|R:C_R(r)|}.$$

$$\mathrm{Pr}(R)\ge$$

$$=$$

$$\frac{1}{|[R,R]|}\left(1+\frac{|[R,R]|-1}{|R:Z(R)|}\right)\ge \frac{|Z(R)|}{|R|}+\frac{p(|R|-|Z(R)|)}{|R||R|}$$

$$\frac{1}{|K(R,R)|}\left(1+\frac{|K(R,R)|-1}{|R:Z(R)|}\right)\ge \frac{1}{|[R,R]|}\left(1+\frac{|[R,R]|-1}{|R:Z(R)|}\right)$$

Full text

*Department of Mathematical Sciences, Tezpur University,

Napaam-784028, Sonitpur, Assam, India.

Emails:  jutirekhadutta@yahoo.com and dbasnet@tezu.ernet.in

Key words: finite ring, commuting probability.

2010 Mathematics Subject Classification: 16U70, 16U80.

1 Introduction

Throughout the paper $R$ denotes a finite ring. The commuting probability of $R$, denoted by $\Pr(R)$, is the probability that a randomly chosen pair of elements of $R$ commute. That is

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The study of $\Pr(R)$ was initiated by MacHale [5] in the year 1976. After the works of Erd$\ddot{\rm o}$s and Tur$\acute{\rm a}$n [4], many papers have been written on commuting probability of finite groups in the last few decades, for example see [3] and the references therein. However, people did not work much on commuting probability of finite rings. We have only few papers [1, 2, 5] on $\Pr(R)$ in the literature. In this paper, we obtain some bounds for $\Pr(R)$.

Recall that $[s,r]$ to denote the additive commutator $sr-rs$ for any two elements $s,r\in R$. By $K(R,R)$ we denote the set $\{[s,r]:s,r\in R\}$ and $[R,R]$ denotes the subgroup of $(R,+)$ generated by $K(R,R)$. Note that $[R,R]$ is the commutator subgroup of $(R,+)$ (see [2]). Also, for any $x\in R$, we write $[x,R]$ to denote the subgroup of $(R,+)$ consisting of all elements of the form $[x,y]$ where $y\in R$.

2 Main Results

Let $C_{R}(r)$ denote the subset $\{s\in R:sr=rs\}$ of $R$, where $r$ is an element of $R$. Then $C_{R}(r)$ is a subring of $R$ known as centralizer of $r$ in $R$. Note that the center $Z(R)$ of $R$ is the intersection of all the centralizers in $R$.

By (1.1), we have

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and hence

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If $p$ is the smallest prime dividing the order of a finite non-commutative ring $R$ then, by [5, Theorem 2], we have

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In the following theorem, we give two bounds for $\Pr(R)$. We shall see that the upper bound for $\Pr(R)$ in Theorem 2.1 is better than (2.2).

Theorem 2.1**.**

Let $R$ be a finite non-commutative ring. If $p$ is the smallest prime dividing $|R|$ then

1. (a)

$\Pr(R)\geq\frac{|Z(R)|}{|R|}+\frac{p(|R|-|Z(R)|)}{|R|^{2}}$* with equality if and only if $|C_{R}(r)|=p$ for all $r\notin Z(R)$.* 2. (b)

$\Pr(R)\leq\frac{(p-1)|Z(R)|+|R|}{p|R|}$* with equality if and only if $|R:C_{R}(r)|=p$ for all $r\notin Z(R)$.*

Proof.

By (2.1), we have

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(a) If $r\notin Z(R)$ then $|C_{R}(r)|\geq p$. Therefore

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with equality if and only if $|C_{R}(r)|=p$ for all $r\notin Z(R)$. Hence, the result follows from (2.3).

(b) If $r\notin Z(R)$ then $|C_{R}(r)|\leq\frac{|R|}{p}$. Therefore

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with equality if and only if $|R:C_{R}(r)|=p$ for all $r\notin Z(R)$. Hence, the result follows from (2.3). ∎

If $R$ is a non-commutative ring and $p$ the smallest prime dividing $|R|$ then $|R:Z(R)|\geq p^{2}$. Therefore

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Thus the bound obtained in Theorem 2.1(b) is better than (2.2).

If $S$ is a subring of $R$ then MacHale [5, Theorem 4] showed that

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We now proceed to derive an improvement of (2.4). We require the following lemma.

Lemma 2.2**.**

Let $N$ be an ideal of a finite non-commutative ring $R$. Then

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The equality holds if $N\cap[R,R]=\{0\}$.

Proof.

For any element $s\in C_{R}(x)+N$, where $s=r+n$ for some $r\in C_{R}(x)$ and $n\in N$, we have $s+N=r+N\in R/N$. Also,

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as $r\in C_{R}(x)$. This proves the first part.

Let $N\cap[R,R]=\{0\}$ and $y+N\in C_{R/N}(x+N)$. Then $y\in R$ and $(y+N)(x+N)=(x+N)(y+N)$. This gives $yx-xy\in N\cap[R,R]=\{0\}$ and so $y\in C_{R}(x)$. Therefore, $y+N\in\frac{C_{R}(x)+N}{N}$. Hence the equality holds. ∎

The following result which is an improvement of (2.4) also gives a relation between $\Pr(R),\Pr(R/N)$ and $\Pr(N)$, where $N$ is an ideal of $R$.

Theorem 2.3**.**

Let $N$ be an ideal of a finite non-commutative finite ring $R$. Then

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The equality holds if $N\cap[R,R]=\{0\}$.

Proof.

We have that

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Let $a+N=S$ where $a\in R\setminus N$. If $C_{R}(n)\cap S=\phi$ then $|C_{R}(n)\cap S|<|C_{N}(n)|$. If $C_{R}(n)\cap S\neq\phi$ then there exists $x_{0}\in C_{R}(n)\cap S$ such that $x_{0}=a+n_{0}$ for some $a\in R\setminus N$ and $n_{0}\in N$. Therefore $x_{0}+N=a+N=S$ and so $S\cap C_{R}(n)=(x_{0}+N)\cap(x_{0}+C_{R}(n))=x_{0}+(N\cap C_{R}(n))=x_{0}+C_{N}(n)$. Hence $|S\cap C_{R}(n)|\leq|C_{N}(n)|$. This gives

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Hence the bound follows.

Let $N\cap[R,R]=\{0\}$. Then, by Lemma 2.2, we have

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If $S=a+N$ then it can be seen that $a+n\in C_{R}(n)\cap S$ for all $n\in N$. Therefore, $C_{R}(n)\cap S\neq\phi$ for all $n\in N$ and for all $S\in R/N$. Thus all the inequalities above become equalities if $N\cap[R,R]=\{0\}$. This completes the proof. ∎

In Lemma 2.3 of [2], Buckley et al. showed that

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The following two results give some improvements of (2.5).

Theorem 2.4**.**

Let $R$ be a finite ring $R$. Then

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with equality if and only if $|K(R,R)|=|[r,R]|$ for all $r\in R\setminus Z(R)$. In particular, if $R$ is non-commutative then $\Pr(R)>\frac{1}{|K(R,R)|}$.

Proof.

By (2.1), we have

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Since $|K(R,R)|\geq|[r,R]|=|R:C_{R}(r)|$ for all $r\in R\setminus Z(R)$, we have

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from which the result follows. ∎

We also have the following lower bound.

Theorem 2.5**.**

Let $S$ be a subring of a finite ring $R$. Then

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with equality if and only if $|[R,R]|=|[r,R]|$ for all $r\in R\setminus Z(R)$. In particular, if $R$ is non-commutative then $\Pr(R)>\frac{1}{|[R,R]|}$.

Proof.

By (2.1), we have

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Since $|[R,R]|\geq|R:C_{R}(r)|$ for all $r\in R\setminus Z(R)$, we have

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from which the result follows. ∎

Let $p$ be the smallest prime dividing $|R|$. If $R$ is non-commutative and $[R,R]\neq R$ then it is easy to see that

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with equality if and only if $|R:[R,R]|=p$. Also,

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with equality if and only if $K(R,R)=[R,R]$. Hence, the lower bound obtained in Theorem 2.4 is better than the lower bounds obtained in Theorem 2.1 and Theorem 2.5. We conclude the paper noting that Theorem 2.4 and Theorem 2.5 are analogous to [7, Theorem A] and [6, Theorem 1] respectively.