Commutativity conditions on rings
نویسندگان
چکیده
منابع مشابه
On Commutativity of Semiperiodic Rings
Let R be a ring with center Z, Jacobson radical J , and set N of all nilpotent elements. Call R semiperiodic if for each x ∈ R\ (J ∪Z), there exist positive integers m, n of opposite parity such that x − x ∈ N . We investigate commutativity of semiperiodic rings, and we provide noncommutative examples. Mathematics Subject Classification (2000). 16U80.
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In this paper, we use the structure theory to prove an analog to a well-known theorem of Herstein as follows: Let R be a ring with center C such that for all x,y ? R either [x,y]= 0 or x-x [x,y]? C for some non negative integer n= n(x,y) dependingon x and y. Then R is commutative.
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Introduction. A celebrated theorem of N. Jacobson [7] asserts that if (1) x*(x) =x for every x in a ring R, where n(x) is an integer greater than one, then R is commutative. In a recent paper [2], I. N. Herstein has shown that it is enough to require that (1) holds for those x in R which are commutators: x= [y, z]=yz — zy of two elements of R. The purpose of this note is to show that if R has n...
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متن کاملGeneralized J-Rings and Commutativity
A J-ring is a ring R with the property that for every x in R there exists an integer n(x)>1 such that x x x n = ) ( , and a well-known theorem of Jacobson states that a Jring is necessarily commutative. With this as motivation, we define a generalized Jring to be a ring R with the property that for all x, y in R0 there exists integers 1 ) ( , 1 ) ( > = > = y m m x n n such that m n xy y x − is ...
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ژورنال
عنوان ژورنال: Bulletin of the Australian Mathematical Society
سال: 1991
ISSN: 0004-9727,1755-1633
DOI: 10.1017/s0004972700029440