Semi-Baer and Semi-Quasi Baer Properties of Skew Generalized Power Series Rings

Document Type : Novel Research Articles

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1 Mathematics Department, Faculty of Science, Assiut university, Assiut, Egypt.

2 Mathematics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt

Abstract

Let R be a ring with identity, (S,≤) an ordered monoid, ω:S→End(R) a monoid homomorphism, and A=R[[S,ω]] the ring of skew generalized power series. The concepts of semi-Baer and semi-quasi Baer rings were introduced by Waphare and Khairnar as extensions of Baer and quasi-Baer rings, respectively. A ring R is called a semi-Baer (semi-quasi Baer) ring if the right annihilator of every subset (right ideal) of R is generated by a multiplicatively finite element in R. In this paper, we examine the behavior of a skew generalized power series ring over a semi-Baer (semi-quasi Baer) ring and prove that, under specific conditions, the ring A is semi-Baer (semi-quasi Baer) if and only if R is semi-Baer (semi-quasi Baer). Also, we prove that if f is a multiplicative finite element of A, then f (1) is a multiplicative finite element of R and determine the conditions under which f = c_(f(1)).

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