The Study On Some Properties Of Monoid Rings

Recently, monoid rings have always been research hotspots in Abstract Algebra. It mainly contains the following research directions. Firstly, whether the extension of some class of rings relative to monoid rings is this class of rings; Secondly, it is studied for some properties and structures of Ore extension rings with an endomorphism or a derivation. This paper researches a-skew M-Armendariz rings, a-skew M-McCoy rings and investigate the strongly a-semicommutativity、strongly a-reversibility and strongly a-symmetric on monoid rings. This thesis mainly consists of the following components:Chapter 1、We introduce the background, development process and research s-tatus of monoid rings, and briefly summarize some important work and results in the literature.Chapter 2、We introduce some essential concepts, such as a-skew M-Armendariz rings, a-skew M-McCoy rings, Baer rings, p.p.-rings, zip rings, a-rigid rings, a-compatible rings, strongly M-a-semicommutative rings, strongly M-a-reversible rings, strongly M-a-symmetric rings, and some results which were frequently used in the sequel.Chapter 3、We mainly investigate the relationship between a-skew M-Armendariz rings,a-skew M-McCoy rings and related rings, and the skew monoid ring extension of Baer rings, p.p.-rings and zip rings. Let M be a monoid, a:M�'End(R) be a homemorphism of a monoid M. The main results are the following:(1) If R is an a-skew M-Armendariz ring, then R is a a-skew N-Armendariz ring, N is a sub-semimonoid of M; (2) If R is an a-skew M-Armendariz ring, then R is a right a-skew M-McCoy ring; (3)Let M be an ordered monoid,I be a reduced ideal of R. If R/I is an a-skew M-Armendariz ring, then R is an a-skew M-Armendariz ring; (4) Let a be an automorphism of monoid, c9= c for each g∈M,c2= c∈R. If R is an a-skew M-Armendariz ring, then R is a Baer ring if and only if R[M;a] is a Baer ring; (5) R is a left a-skew M-McCoy ring if and only if Tn(R) is a left a-skew M-McCoy ring.Chapter 4、This chapter extends the notations of a-semicommutative rings and a-reversible rings in monoid rings, and investigates the relationships between strong-ly M-a-semicommutative rings, strongly M-a-reversible rings and related rings and some extensions. Moreover, we also investigate the relationships between R[M] and nil(R[M]).Chapter 5、We mainly investigate the relationships between strongly M-a-symme tric rings and strongly M-symmetric rings, strongly M-a-semicommutative rings, strong-ly M-a-reversible rings and related rings and some extensions of strongly M-a-symmetric rings and prove that:(1) R is a right strongly M-a-symmetric ring if and only if R is a left strongly M-a-symmetric ring; (2) Let M be an u.p.-monoid, I be an a-ideal of R. If R/I is a strongly M-S-symmetric ring and I is a-rigid, then R is a strongly M-a-symmetric ring; (3)Let a be an automorphism of R. If R is a right Ore ring, then R is a strongly M-a-symmetric ring if and only if Q is a strongly M-α-symmetric ring.Chapter 6、Given a summary about extensions of monoid rings of these classes of rings in this article, we roughly forecasted their possible applications and other research in the future.