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arXiv · 1402.3326

On presimplifiable group rings

Omar Al-mallah
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Abstract

A ring A is called presimplifiable if whenever a; b belongs to A and a = ab, then either a = 0 or b is a unit in A. Let A be a commutative ring and G be an abelian torsion group. For the group ring A[G], we prove that A[G] is presimplifiable if and only if A is presimplifiable and G is a p-group with p belongs to the Jacobson radical of A, and it is shown that A[G] is domainlike (i.e all zero divisors are nilpotents) if and only if A is domainlike and G is a p-group and p is a nilpotent in A. Furthermore, whenever the group ring A[G] is presimplifiable we prove that A[H] is presimplifiable for any subgroup H of G. Also, for a torsion free group G we prove that A[G] is domainlike if and only if A[G] is integral domain.

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Omar Al-mallah. 2014-02-13. On presimplifiable group rings. https://arxiv.org/abs/1402.3326

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