On the sum of annihilators in Monoid rings

A given ring R, is called a left IN-ring if the right annihilator of the intersection of any two left ideals is equal to the sum of their right annihilators. Also, R is said to be a right SA-ring if the sum of the right annihilators of any two ideals forms a right annihilator of an ideal itself. For example, a domain is left Ore if and only if it is left IN. In this paper, our investigation focuses on understanding how the behavior of left IN-rings or right SA-rings relates to monoid rings, and whether these properties transfer between the base ring R and its monoid ring R[M]. Among various findings, for instance, we show that if R[M] is a right SA-ring, then R is also a right SA-ring, and conversely holds true for a semiprime ring R and a unique product monoid M. Additionally, we examine and clarify the connections between these classes of rings and well-known classes of rings.