Commutativity Peroperty on Strongly Periodic Rings

Subject Areas : Algebra

saeed Nasirifar ¹ , shaabanali Safari sabet ²

1 - گروه ریاضی، دانشگاه آزاد اسلامی، واحد تهران مرکزی، تهران، ایران
2 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran

Received: 2022-03-11 Accepted : 2022-10-01 Published : 2024-01-18

Keywords: جابه جایی, حلقه های متناوب, شرط چاکرون, حلقه های قویاً متناوب, رادیکال جیکوبسن,

Abstract :

Let R be an associative ring but not neccessarily unital, denoting the center of the ring R by Z, the radical Jacobsen denoting it by J, and the set of all nilpotent elements of the ring R by N. Ring R is said to be periodic if for every element x∈R there exist distinct m ,n ∈Z^+ such that x^(n )=x^m .The main origin of the definition of the periodic ring, goes back to Jacobsen's theorem.In this theorem he proved that each ring with the condition x=x^(n(x)) ,n(x)>1 ,is commutative.Also,great contemporary mathematicians such as Adilyagqub , Howard.E.Bell, M.Chacron,etc.have worked in this field at some point in their scientific life. For the first time in this article, we have described strongly periodic rings and examined their properties and structure. Ring R is said to be strongly periodic if for every element x∈R \ (J ∪ N) there are positive integers m ,n of opposite parity such that x^(n )-x^m∈N. In this paper, we provide examples of strongly periodic and non-commutative unital rings, and in Theorem 3-6, we also show that the strongly periodic unital ring is commutative or (R, +) a 2-group and R is periodic.

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فهرست منابع

M. Chacron. On a theorem of Herstein. Canada. Journal of mathematics, 21 (1969) 1348-1353.

H.E.Bell.On commutativity of semiperiodic rings.Results in mathematics,53,(2009),19-26.

I.N.Herstein,A generalization of a theorem of Jacobson III.Amer.J.Math.75,(1953),105-111.

Hungerford,W.T.,1973,Algebra. Math . Assoc. of America.