Narrow Results

A ring $R$ is called a left $PS$-ring if its left socle, $\mathrm{\text{Soc}}{(}_{R}R)$, is projective. Equivalently, $R$ is left $PS$-ring if the right annihilator of every maximal left ideal is of the form $eR$, 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 $PS$-ring. Examples to illustrate and delimit the theory are provided.

In this paper, we continue to study the differential inverse power series ring $R[[x^{-1};\delta ]]$, where $R$ is a ring equipped with a derivation $\delta$. We characterize when $R[[x^{-1};\delta ]]$ is a local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, right stable range one, abelian, projective-free, $I$-ring, respectively. Furthermore, we prove that $R[[x^{-1};\delta ]]$ is a domain satisfying the $ACC$ on principal left ideals if and only if so does $R$. Also, for a piecewise prime ring (PWP) $R$ we determine a large class of the differential inverse power series ring $R[[x^{-1};\delta ]]$ which have a generalized triangular matrix representation for which the diagonal rings are prime. In particular, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does $R[[x^{-1};\delta ]]$. Our results extend and unify many existing results.

In the present note, we continue the study of skew inverse Laurent series ring $R((x^{-1};\alpha ,\delta ))$ and skew inverse power series ring $R[[x^{-1};\alpha ,\delta ]]$, where $R$ is a ring equipped with an automorphism $\alpha$ and an $\alpha$-derivation $\delta$. Necessary and sufficient conditions are obtained for $R[[x^{-1};\alpha ,\delta ]]$ to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, projective-free and $I$-ring, respectively. It is shown here that $R((x^{-1};\alpha ,\delta ))$ (respectively $R[[x^{-1};\alpha ,\delta ]]$) is a domain satisfying the ascending chain condition (Acc) on principal left (respectively right) ideals if and only if so does $R$. Also, we investigate the problem when a skew inverse Laurent series ring $R((x^{-1};\alpha ,\delta ))$ has the same Goldie rank as the ring $R$ and is proved that, if $R$ is a semiprime right Goldie ring, then $R((x^{-1};\alpha ,\delta ))$ is semiprimitive. Furthermore, we study on the relationship between the simplicity, semiprimeness, quasi-Baerness and Baerness property of a ring $R$ and these of the skew inverse Laurent series ring. Finally, we consider the problem of determining when $f(x)\in R((x^{-1};\alpha ,\delta ))$ is nilpotent.

In this paper, we prove a necessary and sufficient condition on a ring $R$ with a derivation $\delta$ for the differential polynomial ring $R[x;\delta ]$ to be right quasi-duo. More precisely, $R[x;\delta ]$ is right quasi-duo if and only if $R/J_0$ is commutative and $\delta (R)\subseteq J_0$ where $J_0$ is the intersection of $R$ with the Jacobson radical of $R[x;\delta ]$. The proof is simple and corrects the original proof by Bien and Oinert.

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.