A. Moussavi

Ore Extensions of Skew Armendariz Rings

Let α be an endomorphism and δ an α-derivation of a ring R. We introduce the notion of skew-Armendariz rings which are a generalization of α-skew Armendariz rings and α-rigid rings and extend the classes of non reduced skew-Armendariz rings. Some properties of this generalization are established, and connections of properties of a skew-Armendariz ring R with those of the Ore extension R[x; α, δ] are investigated. As a consequence we extend and unify several known results related to Armendariz rings.

The McCoy Condition on Ore Extensions

Nielsen [29] proved that all reversible rings are McCoy and gave an example of a semicommutative ring that is not right McCoy. When R is a reversible ring with an (α, δ)-condition, namely (α, δ)-compatibility, we observe that R satisfies a McCoy-type property, in the context of Ore extension R[x; α, δ], and provide rich classes of reversible (semicommutative) (α, δ)-compatible rings. It is also shown that semicommutative α-compatible rings are linearly α-skew McCoy and that linearly α-skew McCoy rings are Dedekind finite. Moreover, several extensions of skew McCoy rings and the zip property of these rings are studied.

ON WEAKLY RIGID RINGS

Let R be a ring with a monomorphism α and an α-derivation δ. We introduce (α, δ)-weakly rigid rings which are a generalisation of α-rigid rings and investigate their properties. Every prime ring R is (α, δ)-weakly rigid for any automorphism α and α-derivation δ. It is proved that for any n , a ring R is (α, δ)-weakly rigid if and only if the n -by- n upper triangular matrix ring T n ( R ) is ( , )-weakly rigid if and only if M n ( R ) is ( , )-weakly rigid. Moreover, various classes of (α, δ)-weakly rigid rings is constructed, and several known results are extended. We show that for an (α, δ)-weakly rigid ring R , and the extensions R [ x ], R [[ x ]], R [ x ; α, δ], R [ x , x −1 ; α], R [[ x ; α]], R [[ x , x −1 ; α]], the ring R is quasi-Baer if and only if the extension over R is quasi-Baer. It is also proved that for an (α, δ)-weakly rigid ring R , if any one of the rings R , R [ x ], R [ x ; α, δ] and R [ x , x −1 ; α] is left principally quasi-Baer, then so are the other three. Examples to illustrate and delimit the theory are provided.

Generalized Quasi-Baer Rings

ABSTRACT We say a ring with identity is a generalized right (principally) quasi-Baer if for any (principal) right ideal I of R, the right annihilator of In is generated by an idempotent for some positive integer n, depending on I. The behavior of the generalized right (principally) quasi-Baer condition is investigated with respect to various constructions and extensions. The class of generalized right (principally) quasi-Baer rings includes the right (principally) quasi-Baer rings and is closed under direct product and also under some kinds of upper triangular matrix rings. The generalized right (principally) quasi-Baer condition is a Morita invariant property. Examples to illustrate and delimit the theory are provided.

On Rings Having McCoy-Like Conditions

In [41], Nielsen proves that all reversible rings are McCoy and gives an example of a semicommutative ring that is not right McCoy. At the same time, he also shows that semicommutative rings do have a property close to the McCoy condition. In this article we study weak McCoy rings as a common generalization of McCoy rings and weak Armendariz rings. Relations between the weak McCoy property and other standard ring theoretic properties is considered. We also study the weak skew McCoy condition, a generalization of the standard weak McCoy condition from polynomials to skew polynomial rings. We resolve the structure of weak skew McCoy rings and obtain various necessary or sufficient conditions for a ring to be weak skew McCoy, unifying and generalizing a number of known McCoy-like conditions in the special cases. Constructing various examples, we classify how the weak McCoy property behaves under various ring extensions. As a consequence we extend and unify several known results related to McCoy rings and Armendariz rings [6, 35, 38, 43, 49].

ON ORE EXTENSIONS OF QUASI-BAER RINGS

A ring R is called (right principally) quasi-Baer if the right annihilator of every (principal right) ideal of R is generated by an idempotent. We study on the relationship between the quasi-Baer and p.q.-Baer property of a ring R and these of the Ore extension R[x; α, δ] for any automorphism α and α-derivation δ of R.

Baer and Quasi-Baer Differential Polynomial Rings

A ring R with a derivation δ is called δ-quasi Baer (resp. quasi-Baer), if the right annihilator of every δ-ideal (resp. ideal) of R is generated by an idempotent, as a right ideal. We show the left-right symmetry of δ-(quasi) Baer condition and prove that a ring R is δ-quasi Baer if and only if R[x;δ] is quasi Baer if and only if R[x;δ] is -quasi Baer for every extended derivation of δ. When R is a ring with IFP, then R is δ-Baer if and only if R[x;δ] is Baer if and only if R[x;δ] is -Baer for every extended derivation of δ. A rich source of examples for δ-(quasi) Baer rings is provided.

On Monoid Rings Over Nil Armendariz Ring

Given a ring R and a monoid M, we study the concept of so called nil-Armendariz ring relative to a monoid, which is a common generalization of nil-Armendariz rings and Armendariz rings relative to a monoid. It is done by considering the nil-Armendariz condition on a monoid ring R[M] instead of the polynomial ring R[x]. We prove that several properties transfer between R and the monoid ring R[M], in case R is nil M-Armendariz ring. We resolve the structure of nil M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil M-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the special cases. In particular, we prove that every NI-ring is nil M-Armendariz, for any unique product monoid M. We also classify which of the standard nilpotence properties on polynomial rings pass to monoid rings. We provide various examples and classify how the nil M-Armendariz rings behaves under various ring extensions.

Principally Quasi-Baer Skew Power Series Rings

Let α be an endomorphism of R which is not assumed to be surjective and R be α-compatible. It is shown that the skew power series ring R[[x; α]] is right p.q.-Baer if and only if the skew Laurent series ring R[[x, x −1; α]] is right p.q.-Baer if and only if R is right p.q.-Baer and every countable subset of right semicentral idempotents has a generalized countable join. Examples to illustrate and delimit the theory are provided.

On Skew Quasi-Baer Rings

A ring R with an automorphism α and an α-derivation δ is called (α,δ)-quasi-Baer (resp., quasi-Baer) if the right annihilator of every (α,δ)-ideal (resp. ideal) of R is generated by an idempotent, as a right ideal. We show the left-right symmetry of (α, δ)-quasi Baer condition and prove that a ring R is (α, δ)-quasi Baer if and only if R[x; α, δ] is α-quasi Baer if and only if R[x; α, δ] is -quasi Baer for every extended derivation of δ. When R is a ring with IFP, then R is (α, δ)-Baer if and only if R[x; α, δ] is α-Baer if and only if R[x; α, δ] is -Baer for every extended α-derivation on R[x; α, δ] of δ. A rich source of examples for (α, δ)-quasi Baer rings is provided.