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Let R be an arbitrary ring equipped with an automorphism α and an α-derivation δ. In this paper, we study on the relationship between the piecewise prime property of a ring R and these of the skew inverse Laurent series ring R((x−1; α,δ)) and related rings. Furthermore, for a piecewise prime ring, we determine a large class of rings which have a generalized triangular matrix representation for which the diagonal rings are prime. Our results extend and unify many available results. Examples for illustration and delimitation of the theory are provided. [ABSTRACT FROM AUTHOR]

In this paper we continue the study of various ring theoretic properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangular matrix ring to satisfy a certain ring property which is among being Kasch, completely primary, quasi-duo, 2-primal, NI, semiprimitive, projective-free, etc. We also characterize when a general Morita context is weakly principally quasi-Baer or strongly right mininjective. [ABSTRACT FROM AUTHOR]

AbstractLet R be a ring with a monomorphism α and an α-derivation δ. In this article, we give a simple and different proof about the semiprimitivity of Ore extensions which states that the skew polynomial ring R[x;α,δ] is semiprimitive reduced if and only if R is α-rigid. This unifies and extends a number of known results on the Jacobson radical in the special cases. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of rings whose the set of all nilpotent elements has special conditions. Important examples are provided to illustrate the applications of the results. [ABSTRACT FROM AUTHOR]

In this paper, we continue to study skew inverse Laurent series ring R((x−1; α,δ)), where R is a ring equipped with an automorphism α and an α-derivation δ. We directly prove that R((x−1; α,δ)) is semiprimitive reduced if and only if R is α-rigid. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of R((x−1; α,δ)) whose set of all nilpotent elements has special conditions. Moreover, necessary and sufficient conditions are obtained for the skew inverse Laurent series ring to satisfy a certain ring property which is among being right Artinian, completely primary, right perfect, (semi)local, semiperfect, semiprimary, semiregular, semisimple and strongly regular, respectively. [ABSTRACT FROM AUTHOR]

A ring R is called a left P S -ring if its left socle, Soc (R R) , is projective. Equivalently, R is left P S -ring if the right annihilator of every maximal left ideal is of the form e R , where e is an idempotent in R. In this paper, we characterize when a (trivial) Morita context or a generalized triangular matrix ring is a left P S -ring. Examples to illustrate and delimit the theory are provided. [ABSTRACT FROM AUTHOR]

Let R be an associative ring equipped with an automorphism α and an α -derivation δ. In this note, we characterize when a skew inverse Laurent series ring R ((x − 1 ; α , δ)) and a skew inverse power series ring R [ [ x − 1 ; α , δ ] ] are 2-primal, and we obtain partial characterizations for those to be NI. [ABSTRACT FROM AUTHOR]

The aim of our paper is to study the primitivity of the skew inverse Laurent series rings R ((x − 1 ; α , δ)) and the skew Laurent power series rings R [ [ x , x − 1 ; α ] ] , where R is an associative ring equipped with an automorphism α and an α -derivation δ. Examples to illustrate and delimit the theory are provided. [ABSTRACT FROM AUTHOR]

A generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of differential inverse power series type (or simply, D I P S -quasi-Armendariz), is introduced and studied. It is shown that the D I P S -quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. Various classes of non-semiprime D I P S -quasi-Armendariz rings are provided, and a number of properties of this generalization are established. Some characterizations for the differential inverse power series ring R [ [ x − 1 ;   δ ] ] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP are concluded, where δ is a derivation on the ring R. Finally, miscellaneous examples to illustrate and delimit the theory are given. [ABSTRACT FROM AUTHOR]

In the present note, we continue the study of skew inverse Laurent series ring and skew inverse power series ring , where is a ring equipped with an automorphism and an -derivation . Necessary and sufficient conditions are obtained for to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, projective-free and -ring, respectively. It is shown here that (respectively ) is a domain satisfying the ascending chain condition (Acc) on principal left (respectively right) ideals if and only if so does . Also, we investigate the problem when a skew inverse Laurent series ring has the same Goldie rank as the ring and is proved that, if is a semiprime right Goldie ring, then is semiprimitive. Furthermore, we study on the relationship between the simplicity, semiprimeness, quasi-Baerness and Baerness property of a ring and these of the skew inverse Laurent series ring. Finally, we consider the problem of determining when is nilpotent. [ABSTRACT FROM AUTHOR]

Let be a ring, a strictly ordered monoid and a monoid homomorphism. The skew generalized power series ring is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we consider the problem of determining when is nilpotent in . We study various annihilator properties and a variety of conditions and related properties that the skew generalized power series inherits from . We also introduce and study the -McCoy condition on , a generalization of the standard McCoy condition from polynomials to skew generalized power series. We resolve the structure of -McCoy rings and obtain various necessary or sufficient conditions for a ring to be -McCoy. As particular cases of our general results we obtain several new theorems on the McCoy condition. Moreover various examples of -McCoy rings are provided. [ABSTRACT FROM AUTHOR]