On commuting probability of finite rings

Dutta, Jutirekha; Basnet, Dhiren Kumar; Nath, Rajat Kanti

doi:10.1016/j.indag.2016.10.002

Cited by 13 publications

(23 citation statements)

References 17 publications

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“…We shall conclude this section by the following theorem which is an improvement of Theorem 2.13 of [5]. Thus, the first part of the result follows.…”

Section: -9mentioning

confidence: 68%

“…It is worth mentioning that the second part of Theorem 2.4 in [5] follows from the first part of Proposition 3.5. Proposition 3.6.…”

Section: -5mentioning

confidence: 94%

“…It is worth noting that Theorem 2.17 of [5] follows from Proposition 3.7. The last part follows considering p = 2.…”

Section: In Particular If Z(s K) = S Then Pr(s K) > 1 |[Sk]|mentioning

confidence: 97%

“…[4] and the references therein), but somehow people have neglected the commuting probability of finite rings. At this moment, we have very few papers [1,2,5,6,12] in the literature on the commuting probability of finite rings. Let S and K be two additive subgroups of R and r ∈ R. We define Pr r (S, K) in the following way:…”

Section: Introductionmentioning

confidence: 99%

“…where [s, k] := sk −ks is the additive commutator of s and k. Thus, Pr r (S, K) is the probability that the additive commutator of a randomly chosen pair of elements, one from S and the other from K, is equal to a given element r of R. The case when S = K = R and the case when S is a subring and K = R are considered in [7] and [8], respectively. If r = 0, then we write Pr r (S, K) = Pr(S, K) = |{(s, k) ∈ S × K : sk = ks}| |S × K| .…”

Section: Introductionmentioning

confidence: 99%

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A generalization of commuting probability of finite rings

Dutta

Nath

2018

Asian-European J. Math.

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The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different additive subgroups of a finite noncommutative ring equals a given element of that ring. We obtain several results on this probability including a computing formula, some bounds and characterizations. Proposition 2.1. Let S and K be two additive subgroups of R and r ∈ [S, K]. Then, Pr r (S, K) = Pr −r (K, S). However, if 2r = 0, then Pr r (S, K) = Pr r (K, S). 1850023-2 Asian-European J. Math. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 04/23/17. For personal use only. A generalization of commuting probability of finite rings Proof. Let X = {(s, k) ∈ S×K : [s, k] = r} and Y = {(k, s) ∈ K ×S : [k, s] = −r}.It is easy to see that (s, k) → (k, s) defines a bijective mapping from X to Y . Therefore, |X| = |Y | and the result follows.Second part follows from the fact that r = −r if 2r = 0.Proposition 2.2. Let S i and K i be two additive subgroups of finite noncommutative rings R i for i = 1, 2, respectively. If (r 1 , r 2 ) ∈ R 1 × R 2 , then s 2 ), (k 1 , k 2 )] = (r 1 , r 2 )}. 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. 1850023-3Asian-European J. Math. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 04/23/17. For personal use only.

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“…We shall conclude this section by the following theorem which is an improvement of Theorem 2.13 of [5]. Thus, the first part of the result follows.…”

Section: -9mentioning

confidence: 68%

“…It is worth mentioning that the second part of Theorem 2.4 in [5] follows from the first part of Proposition 3.5. Proposition 3.6.…”

Section: -5mentioning

confidence: 94%

“…It is worth noting that Theorem 2.17 of [5] follows from Proposition 3.7. The last part follows considering p = 2.…”

Section: In Particular If Z(s K) = S Then Pr(s K) > 1 |[Sk]|mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%