Narrow Results

We continue the study of the reflexivity of ideals, introduced by Mason, and extend this notion to the subsets in rings. We first construct the smallest reflexive ideal containing $S$ from any proper ideal $S$ of any given ring $R$; by which we can construct reflexive ideals but not semiprime in a kind of noncommutative ring. A subset $T$ of a ring $R$ is called reflexively closed if $aRb\subseteq T$ for $a,b\in R$ implies $bRa\subseteq T$, checking that a ring $R$ is symmetric if and only if the right (left) annihilator of $a$ is reflexively closed for any $a\in R$. We prove that the set of all nilpotent elements in a ring $R$ is reflexively closed if and only if $aR$ is nil for any nilpotent element $a$ in $R$; and that the Köthe’s conjecture holds if and only if the union (sum) of the upper nilradical and any nil right ideal is reflexively closed. We provide another process to show that the set of all nilpotent elements of the polynomial ring over an NI ring need not be reflexively closed.

Jacobson investigated the structure of rings with the property that some power of each element is central, in the procedure of the study of commutativity. In this paper, we consider a class of rings in which this property occurs only for central elements, and such rings are called ppnc. We first prove that a noncommutative ppnc ring is infinite, and that a ppnc ring is commutative when it is a K-ring or a locally finite ring. We next study the structure of ppnc rings, and the relation between ppnc rings and related concepts (for example, K-rings, commutative rings and NI rings), through matrix rings, polynomial rings, right quotient rings and direct products.

In this paper, we continue to study skew inverse Laurent series ring $R((x^{-1};\alpha ,\delta ))$, where $R$ is a ring equipped with an automorphism $\alpha$ and an $\alpha$-derivation $\delta$. We directly prove that $R((x^{-1};\alpha ,\delta ))$ is semiprimitive reduced if and only if $R$ is $\alpha$-rigid. Also, as an application of our results, by imposing constraints on $\alpha$ and $\delta$, we completely identify the Jacobson radical of $R((x^{-1};\alpha ,\delta ))$ 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.

It is proved that for matrices $A$, $B$ in the $n$ by $n$ upper triangular matrix ring $T_n(R)$ over a domain $R$, if $AB$ is nonzero and central in $T_n(R)$ then $AB=BA$. The $n$ by $n$ full matrix rings over right Noetherian domains are also shown to have this property. In this article we treat a ring property that is a generalization of this result, and a ring with such a property is said to be weakly reversible-over-center. The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains. The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally. We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.

Marks called a ring NI if the upper nilradical contains all nilpotent elements. Smoktunowicz constructed an NI ring over which the polynomial ring is not NI. We study some conditions under which polynomial rings are NI. In the process we call a ring Rpolynomial-NI if R[Y] is NI for any finite set Y of commuting indeterminates over R. We show that R is polynomial-NI if and only if R[X] is polynomial-NI for any (possibly infinite) set X of commuting indeterminates over R. It is also shown that R is polynomial-NI if and only if R[x] is NI when R is a ring such that every finite subset of R generates a subring of bounded index of nilpotency, where x is an indeterminate over R.