Generalized Morphic Group Rings

Abstract

Let A be a commutative ring with identity, G be an abelian group, and consider the group ring AG. A ring A is called a generalized morphic ring (GM ring) if the annihilator of each element in A is principal. In this article, we showed that if AG is a GM ring, then so is A. The converse was proved to be false. We try to put some conditions on A or G to get the converse. Among many other results, we showed that if A is an Armendariz ring and G is a torsion free group, then AG is a GM ring if and only if A is. Moreover, if C_m denotes the multiplicative cyclic group of order m and Z the ring of integers modulo n, we justified that the ring Z_m is a GM ring if and only if, whenever p is a prime dividing gcd(n,m), then p² ∤ n. We also proved that for an integral domain D with char(D) = p, the group ring DC_p is a GM ring.

Keywords: Morphic ring; Generalized morphic ring; Morphic group ring; Generalized morphic group ring.

2020 Mathematics Subject Classification. 26A25; 26A35