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On commuting probability of finite rings II

The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different subrings of a finite non-commutative ring equals a given element of that ring. We obtain several results on this probability including a computing formula, some bounds and characterizations.


Introduction
Throughout this paper R denotes a finite non-commutative 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 Pr(R) = |{(r, s) ∈ R × R : [r, s] = 0}| |R × R| where [r, s] := rs − sr is the additive commutator of r and s and 0 is the zero element of R. Clearly Pr(R) = 1 if and only if R is commutative. The study of commuting probability of finite rings was initiated by MacHale [10] in the year 1976 motivated by the commuting probability of finite groups. After Erdös [7], many authors have worked on the commuting probability of finite groups and its generalizations (conf. [4] and the references therein) but somehow people have neglected commuting probability of finite rings. At this moment, we have very few papers in the literature on commuting probability of finite rings [1,2,5,6,10]. In this paper, we study a generalization of Pr(R).
Let S and K be two subrings of R and r ∈ R. We define Pr r (S, K) in the following way It may be mentioned here that some connections between Pr(S, K) and generalized non-commuting graph of R can be found in [6]. In [5], Pr(S, R) is studied extensively.
In this paper, we obtain several results on Pr r (S, K) including a computing formula, some bounds and characterizations. The motivation of this paper lies in [3,11,12] where analogous generalizations of commuting probability of finite groups are studied.
For any two subrings S and K, we write [S, K] and [s, K] for s ∈ S to denote the additive subgroups of (R, +) generated by the sets {[s, k] : s ∈ S, k ∈ K} and {[s, k] : k ∈ K} respectively. It can be seen that any element of [s, K] is of the form [s, k] for some k ∈ K and so [s, K] = {[s, k] : k ∈ K}. Let Z(S, K) := {s ∈ S : sk = ks ∀k ∈ K}. Then Z(S, K) = S ∩ Z(K) if S ⊆ K and Z(K, K) = Z(K), the center of K. Also, for r ∈ R the set C S (r) := {s ∈ S : sr = rs} is a subring of S and ∩ r∈K C S (r) = Z(S, K). We write R/S or R S to denote the additive factor group, for any subring S of R, and |R : S| to denote the index of (S, +) in (R, +). The isomorphisms considered in this paper are the additive group isomorphisms. It is easy to see that Pr r (S, K) = 1 if and only if r = 0 and [S, K] = {0}. Also, Pr r (S, K) = 0 if and only if r / ∈ {[s, k] : s ∈ S, k ∈ K}. Therefore, we consider r to be an element of {[s, k] : s ∈ S, k ∈ K} throughout the paper.

Preliminary results
In this section, we deduce some elementary results on Pr r (S, K) and derive a computing formula for Pr r (S, K). We begin with the following result which shows that Pr r (S, K) is not symmetric with respect to S and K. Proposition 2.2. Let S i and K i be two subrings of finite non-commutative rings R i for i = 1, 2 respectively. If (r 1 , r 2 ) ∈ R 1 × R 2 then Then ((s 1 , k 1 ), (s 2 , k 2 )) → ((s 1 , s 2 ), (k 1 , k 2 )) defines a bijective map from X 1 × X 2 to Y . Therefore, |Y | = |X 1 ||X 2 | and hence the result follows.
Initially, it was challenging for us to derive a computing formula for Pr r (S, K) since there is no analogous concept of conjugacy class and no analogous character theoretic results for rings. Finally, we are able to get a formula. The following two lemmas play important role in obtaining our computing formula for Pr r (S, K).
. Hence, the lemma follows. Proof. Let t ∈ T s,r and p ∈ t+C K (s). Then [s, p] = r and so p ∈ T s,r . Therefore, t + C K (s) ⊆ T s,r . Again, if k ∈ T s,r then (k − t) ∈ C K (s) and so k ∈ t + C K (s). Therefore, t + C K (s) ⊆ T s,r . Hence part (a) follows. Part (b) follows from the fact that y ∈ T s,r if and only of r ∈ [s, K].
Now we state and prove our main result of this section.
Theorem 2.5. Let S and K be two subrings of R. Then |C K (s)|. (2.1) The second part follows from (2.1) and Lemma 2.3.
Using Proposition 2.1, we get the following corollary of Theorem 2.5.
Corollary 2.6. Let S and K be two subrings of R. Then It is worth mentioning that Equation (2.1) of [2] and [5] also follow from Corollary 2.6.
We conclude this section by the following two lemmas.
Lemma 2.7. [5, Lemma 2.12] Let H and N be two subrings of a non-commutative ring R such that N is an ideal of R and N ⊆ H. Then

Some bounds and characterizations
In this section, we derive some bounds for Pr r (S, K) and characterizations of subrings S and K in terms of Pr(S, K). We begin with the following lower bounds. Proof. Since r = 0 we have the set C : Proof. By Theorem 2.5 and Corollary 2.6, we have The equality holds if and only if r = 0. If S ⊆ K then we have The equality holds if and only if S = K.
Proposition 3.3. Let S and K be two subrings of R. If p is the smallest prime dividing |R| and r = 0 then Hence the result follows from Theorem 2.5.
Proof. By Theorem 2.5, we have Hence the result follows.

Note that equality holds in Proposition 3.4 if and only if
If r = 0 then the condition of equality reduces to S 1 = S 2 and The equality holds if and only if r = 0 and S 1 = S 2 .
Proof. Putting K 1 = K 2 = R in Proposition 3.4, we get Hence the result follows from Proposition 3.2.
Proposition 3.6. Let S, K 1 and K 2 be three subrings of R.
for all s ∈ S, by Corollary 2.6, we have with equality if and only if [s, By Corollary 2.6, we also have the equality holds if and only if |C S (k)| = 1 for all k ∈ K 2 \ K 1 . Hence the result follows from (3.1) and (3.2).
It is worth mentioning that the second part of [5, Theorem 2.4] follows from the first part of Proposition 3.6. Proof. If s ∈ S \ Z(S, K) then C K (s) = K and hence |K| |CK(s)| ≥ p. Therefore, by Corollary 2.6, we have If S = K then for s ∈ S \ Z(S, K) we have Z(S, K) C K (s) = K and so |C K (s)| ≥ p|Z(S, K)|. Therefore, by Corollary 2.6, we have  Proof. Since [S, K] = {0} we have Z(S, K) = S. Therefore |Z(S, K)| ≤ |S| p . Hence, by second part of Theorem 3.8, we have The particular case follows from the fact p ≥ 2 and 2p−1 p 2 ≤ 3 4 for any prime p.   Proof. If Pr(S, K) = 2p−1 p 2 then, by Corollary 2.6, we have p divides |S||K| and hence p divides |R|.
For the second part we have, by Theorem 3.8, The last part follows considering p = 2.
The following lemma is useful in the subsequent results. Proof. By Corollary 2.6 we have and Hence the lemma follows.
It may be mentioned here that [5, Theorem 2.2] follows from the above lemma.
By second part of Theorem 3.8, we have The particular case follows considering p = 2.
The following proposition gives partial converse of Theorem 3.11 and Theorem 3.14.
If p is the smallest prime dividing |R| and |K : S| = p then |C K (s)| = |K| p for s ∈ S \ Z(S, K) and hence Pr(S, K) = 2p−1 for all s ∈ S \ Z(S, K). Now, by Corollary 2.6 and (3.6), we have If p is the smallest prime dividing |R| and |K : S| = 1 then |C K (s)| ≥ p|Z(S, K)| = |S| p for s ∈ S \ Z(S, K). Also, C K (s) S and so |C K (s)| = |S| p for s ∈ S \ Z(S, K). Therefore, Corollary 2.6 gives Pr(S, K) = 1 The following corollary follows immediately.  We also have the following result.
We shall conclude this section by the following proposition which is an improvement of [5, Theorem 2.13]. Let a+I = P where a ∈ K\I. If C K (u)∩P = φ then |C K (u)∩P | = 0. Therefore, |C K (u) ∩ P | < |C I (u)| as |C I (u)| ≥ 1. On the other hand, if C K (u) ∩ P = φ then there exist x ∈ C K (u) ∩ P and x = a + v for some v ∈ I which implies x + I = a + I = P . Therefore, Thus, first part of the result follows.
If I ∩ [S, R] = 0 then equality holds in Lemma 2.7 and hence equality holds in (3.7). Further, if P = a + I and k ∈ P then k = a + u for some u ∈ I and a ∈ K \ I. Therefore, k ∈ K and u ∈ I ⊆ S and hence [u, k] ∈ [S, K]. Also u ∈ I and k ∈ K ⊆ R gives uk, ku ∈ I and so [u, k] ∈ I. Hence, [u, k] ∈ [S, R] ∩ I and so k ∈ C K (u). Therefore, C K (u) ∩ P = φ and |C K (u) ∩ P | = |C I (u)|. Hence the equality holds.

Isoclinism and commuting probability
The concept of isoclinism between groups was introduced by Hall [8] in 1940. In 1995, Lescot [9] showed that the commuting probabilities of two isoclinic finite groups are same. Recently, Bukley, MacHale and Ni Shé [2] have introduced Z-isoclinism between two rings and showed that the commuting probabilities of two Z-isoclinic finite rings are same. Further, Dutta, Basnet and Nath [6] have generalized the concept of Z-isoclinism between two rings as given in the following definition.
Proof. By Theorem 2.5, we have K1] noting that r ∈ [s 1 , K 1 ] if and only if r ∈ [s 1 + z, K 1 ] and C K1 (s 1 ) = C K1 (s 1 + z) for all z ∈ Z(S 1 , K 1 ). Now, by Lemma 2.3, we have Similarly, it can be seen that Since (φ, ψ) is a Z-isoclinism between (S 1 , K 1 ) and (S 2 , K 2 ) we have We conclude this paper by the following result.