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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| Language: | English |
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Wiley
1991-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171291000911 |
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| author | Hamza A. S. Abujabal |
| author_facet | Hamza A. S. Abujabal |
| author_sort | Hamza A. S. Abujabal |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-242c12120cc4472fa87be63dcadd76c5 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1991-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-242c12120cc4472fa87be63dcadd76c52025-02-03T01:07:34ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251991-01-0114468368810.1155/S0161171291000911Commutativity theorems for rings with constraints on commutatorsHamza A. S. Abujabal0Department of Mathematics, Faculty of Science, King Abdul Aziz University, P. O. Box 31464, Jeddah 21497, Saudi ArabiaIn 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.http://dx.doi.org/10.1155/S0161171291000911commutativty of ringstorsion free ringsring with unity semi-prime rings. |
| spellingShingle | Hamza A. S. Abujabal Commutativity theorems for rings with constraints on commutators International Journal of Mathematics and Mathematical Sciences commutativty of rings torsion free rings ring with unity semi-prime rings. |
| title | Commutativity theorems for rings with constraints on commutators |
| title_full | Commutativity theorems for rings with constraints on commutators |
| title_fullStr | Commutativity theorems for rings with constraints on commutators |
| title_full_unstemmed | Commutativity theorems for rings with constraints on commutators |
| title_short | Commutativity theorems for rings with constraints on commutators |
| title_sort | commutativity theorems for rings with constraints on commutators |
| topic | commutativty of rings torsion free rings ring with unity semi-prime rings. |
| url | http://dx.doi.org/10.1155/S0161171291000911 |
| work_keys_str_mv | AT hamzaasabujabal commutativitytheoremsforringswithconstraintsoncommutators |