Symmetric Rings?~*-Symmetric Rings And ~*-skew Polynomial Rings

In recent years,the general rings(not necessarily with identity)and rings with involution have become the important research objects in algebra.There are three main research directions.Firstly,extending the properties of rings with identity to general rings,and study them.Secondly,for some properties of ring theory,generalizing them to the ring with involution and studying them;Thirdly,investigating the properties and the structures of*-skew polynomial rings.We follow these three methods in this paper,mainly research the relationships between general symmetric rings?*-symmetric rings and their related rings,discuss the extensions of*-symmetric rings and then investigate the properties of*-skew polynomial rings over the commutative rings.This thesis mainly consists of the following parts:Chapter 1:We introduce the background,development process and research status of symmetric rings?*-symmetric rings and*-skew polynomial rings,and briefly summarize some important work and results in the literature.Chapter 2:We introduce some essential concepts and some results.Chapter 3:We mainly introduce the concept of general symmetric rings in the category of general rings(not necessarily with identity),extend the concept of right(left)symmetric rings,discuss the relationships between general symmetric rings and related rings and study their extensions.Chapter 4:We mainly study the relationships between*-symmetric rings and their related rings,and the extensions of*-symmetric rings.The main results are the following:(1)Give some conditions for a*-symmetric ring to be*-reversible.(2)If R is a*-symmetric ring,then some extensions of it are also*-symmetric.(3)If R is a reduced ring,then R is*-symmetric if and only if R[x;*]is*-symmetric.Chapter 5:This chapter we at first study the the*-skew polynomial rings of quasi Baer*-rings and*-right p.q.Baer rings,give the definition of*-right p.q.Baer rings.We prove:(1)If R is a*-reversible quasi Baer*-ring,then R[x;*]is a quasi Baer*-ring.(2)Let*be the proper involution on R and R be*-reversible,if R[x;*]is a quasi Baer*-ring,then R is a quasi Baer*-ring,too.(3)Let*be a proper involution on R and R be a*-right p.q.Baer ring,and for any e E Sl(R)be to meet the conditions:for any r?E R,re=0 can imply to re*=0,then R[x;*]is*-right p.q.Baer ring.After that we research other properties of*-skew polynomial rings and prove that:(4)If is a*-skew Armendariz?*-reversible right zip ring,then R[x;*]is a right zip ring;(5)If R is*-reversible,R[x;*]is a McCoy ring if and only if R is a*-McCoy ring;(6)R is a*-skew Armendariz and Abelian ring with idempotents are all self adjoint,then R[x;*]is a p.p.-ring if and only if R is a p.p.-ringChapter 6:We give a summary about general symmetric rings?*-symmetric rings and*-skew polynomial rings in this article,and make a further outlook on the study of these rings.