Some Properties Of The Skew Triangular Matrix Rings

The skew triangular matrix rings have become an important research object in algebra.It mainly contains the following research directions.Firstly,it is studied for some properties and structures of the skew triangular matrix rings;Secondly,whether the skew triangular matrix rings of some class of rings is this class of rings.This paper is researched on the second direction including the skew triangular matrix rings of Semi-boolean ring,Kasch ring,ring with right stable rank 1,EIFP ring,Hermite ring,?-Armendariz ring,M-Armendariz ring,(?,?)-weakly rigid ring,strong reversible ring.Furthermore,in the special condition,we also study clean?2-good and other properties of T(R,n,?)rings.This paper mainly consists of the following components:In chapter 1,we introduce the background,development process and research status of Tn(R??)rings,and briefly summarize some important work and results in the literature.In chapter 2,we introduce some essential concepts,such as Semi-boolean ring,Kasch ring,ring with right stable rank 1,EIFP ring,Hermite ring,?-Armendariz ring,M-Armendariz ring,(?,?)-weakly rigid ring,strong reversible ring,and some results which were frequently used in the sequel.In chapter 3,we mainly discuss for some properties and structures of the skew triangular matrix rings,and these enrich the theoretical study of the skew triangular matrix rings.The main results are the following:(1)Let a be an endomorphism of a ring R with ?(1)=1.Then R is a Hermite ring if and only if Tn(R,?)is a Hermite ring;(2)A ring R is right weakly McCoy if and only if Tn(R,?)is right weakly McCoy;(3)Let M be a Monoid.If a is rigid,then R is M-Armendariz if and only if Tn(R,?)is M-Armendariz;(4)If ? is rigid,then R is strongly reversible if and only if Tn(R,?)is strongly reversible;(5)Let ? be an endomorphism and ? be an ?-derivation of a ring R.If ? and a are all rigid endomorphism of ring R,??=?? and R is ?-weakly rigid,then R is(?,?)-weakly rigid if and only if Tn(R,?)is(?,?)-weakly rigid.In chapter 4,we introduce some properties of the general skew triangular matrix rings.Let a be an endomorphism of a ring R,and ?(1)=1.we prove that:Let g(x)? C(R)[x],then R is a(k,g(x))-clean ring(resp.,a 2-good ring?a k-good ring)if and only if T(R,n,?)is a(k,g(x))-clean ring(resp.,a 2-good ring?a k-good ring);(2)Let ? is a rigid automorphism of ring R,M is a monoid.If R[M]is a right APP ring,then S(R,n,?)is a M-quasi-Armendariz ring.In chapter 5,we give a summary about properties of Tn(R,?)rings in this article.Moreover,we also roughly forecast other research of the skew triangular matrix rings in the future.