Commutativity theorems for rings with constraints on commutators
In this paper, we generalize some well-known commutativity theorems for associative rings as follows: Let n>1, m, s, and t be fixed non-negative integers such that s≠m−1, or t≠n−1, and let R be a ring with unity 1 satisfying the polynomial identity ys[xn,y]=[x,ym]xt for all y∈R. Suppose that (i)...
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| Format: | Article |
| Language: | English |
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Hindawi Limited
1991-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171291000911 |
| Summary: | In this paper, we generalize some well-known commutativity theorems for
associative rings as follows: Let n>1, m, s, and t be fixed non-negative integers such that
s≠m−1, or t≠n−1, and let R be a ring with unity 1 satisfying the polynomial identity
ys[xn,y]=[x,ym]xt for all y∈R. Suppose that (i) R has Q(n) (that is n[x,y]=0 implies
[x,y]=0); (ii) the set of all nilpotent elements of R is central for t>0, and (iii) the set of
all zero-divisors of R is also central for t>0. Then R is commutative. If Q(n) is replaced by
m and n are relatively prime positive integers, then R is commutative if extra constraint is
given. Other related commutativity results are also obtained. |
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| ISSN: | 0161-1712 1687-0425 |