Commutativity in Finite Rings

MacHale, Des

doi:10.1080/00029890.1976.11994032

Cited by 34 publications

(36 citation statements)

References 2 publications

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“…Thus, Proof. This will follows from the "ΛΓover C" theorem [5, p. 20], which gives us that 9 then Kj{K r C(K f )) can be generated by n elements XtiK'CίK')) with ^ e K. Now we can use the result of P. Hall [5, p. 266] …”

Section: (Where P K -\H'\) In Particular [H: Z(h)] Is a Square Anmentioning

confidence: 99%

“…This tells us that \C(K')\ is at most equal to the number of values the w-tuple {[y, #J, 1 <* i <; n} assumes as y ranges over C(K') 9 which is therefore at most…”

Section: Therefore For Y ί9 Y 2 E C(k')> If [Y ίF X T ] = [Y 2 X Tmentioning

confidence: 99%

“…In the case of 5/8, it is precisely the set of groups G with G' ~ C 2 and G/Z C 2 x C 2 that have this value Pr (G). (Actually, the first constraint is superfluous: see [9].) V* Concluding remarks* There are several open questions relating to Pr ((?).…”

Section: G' < Z(g)mentioning

confidence: 99%

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What is the probability that two elements of a finite group commute?

Rusin¹

1979

Pacific J. Math.

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Section: (Where P K -\H'\) In Particular [H: Z(h)] Is a Square Anmentioning

confidence: 99%

“…This tells us that \C(K')\ is at most equal to the number of values the w-tuple {[y, #J, 1 <* i <; n} assumes as y ranges over C(K') 9 which is therefore at most…”

Section: Therefore For Y ί9 Y 2 E C(k')> If [Y ίF X T ] = [Y 2 X Tmentioning

confidence: 99%

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What is the probability that two elements of a finite group commute?

Rusin¹

1979

Pacific J. Math.

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“…However, it was shown in [9] that Pr(R) is no larger than Pr(S) whenever S is a subring of R. Our first result gives a new proof of this result, one that allows us to characterize when equality occurs. Our second result is similar, but involves a comparison with quotient rings.…”

Section: Introductionmentioning

confidence: 81%

Commuting probability for subrings and quotient rings

Buckley¹,

MacHale²

2017

Journal of Algebra Combinatorics Discrete Structures and Applications

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“…The commutativity for finite rings defined and studied by McHale in 1976 (see [8]). The article [7] studied the case when P r (A) ≥ 1 2 in 1995 for a finite group A.…”

Section: Introductionmentioning

confidence: 99%

On a class of nowhere commutative semigroup rings

Ahmadi¹,

Doostie²

2013

imf

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For a finite algebraic structure A the commutativity degree of A, denoted by P r (A), is the probability that two elements of A commute. Since 1973, P r (A) studied for finite groups showing that P r (A) ≤ 5 8 . The study of "almost commutative" semigroups in 2011 showed that for some finite semigroup A, P r (A) > 5 8 and even P r (A) may be arbitrarily close to 1. Since 0 ≤ P r (A) ≤ 1 then looking for algebraic structures A such that P r (A) → 0, is of interest, for, the centralizer of every element of such A should be a singleton. In this paper for every integer n ≥ 2 and every prime p ≥ 3 we give an infinite class of finite semigroup rings A n,p = Z p (P n ) of order p n where, P n is a non-commutative semigroup and show that P r (A n,p ) → 0, for sufficiently large values of n and p. We name such semigroup rings as "extremely non-commutative" semigroup rings.

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Commutativity in Finite Rings

Cited by 34 publications

References 2 publications

What is the probability that two elements of a finite group commute?

What is the probability that two elements of a finite group commute?

Commuting probability for subrings and quotient rings

On a class of nowhere commutative semigroup rings

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