Some Extensions of Generalized Morphic Rings and EM-rings

Abstract

Let R be a commutative ring with unity. The main objective of this article is to study the relationships between PP-rings, generalized morphic rings and EM-rings. Although PP-rings are included in the later rings, the converse is not in general true. We put necessary and sufficient conditions to ensure the converse using idealization and polynomial rings

$EM-$Graded Rings

“…We also prove that if R is an EM − G-graded ring and S ⊆ h(R) is multiplicatively closed then S −1 R is EM − G−graded ring. In this section we also obtain a nice result related to the idealization R(+)R that generalizes [10,Theorem 5.1]. We show that R is an EM−ring if and only if R(+)R is an EM − Z 2 −graded ring with gradation…”

“…Ganam and Abuosba in [10] proved that if R(+)R (equivalently R[x]/(x 2 )) is an EM−ring then so is R. However the converse of this result is not true. From Theorem 3.5 and Theorem 3.23 we obtain the following result.…”

“…As a generalization, Abuosba and Ghanem [4] introduce the concept of EM−rings as follows: A ring R is called EM−ring if every zero divisor polynomial in R[x] has an annihilating content. These rings were studied extensively in [4] and [10].…”