Michał Ziembowski

On Von Neumann Regular Rings of Skew Generalized Power Series

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboims construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S∖{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]].

Duo, Bézout, and Distributive Rings of Skew Power Series

We give necessary and sufficient conditions on a ring R and an endomorphism σ of R for the skew power series ring R[[x; σ]] to be right duo right Bezout. In particular, we prove that R[[x; σ]] is right duo right Bezout if and only if R[[x; σ]] is reduced right distributive if and only if R[[x; σ]] is right duo of weak dimension less than or equal to 1 if and only if R is N0-injective strongly regular and σ is bijective and idempotent-stabilizing, extending to skew power series rings the Brewer-Rutter-Watkins characterization of commutative B´ezout power series rings.

Derivations and bounded nilpotence index

We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x;δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about locally nilpotent PI rings.

On right McCoy rings and right McCoy rings relative to u.p.-monoids

In this paper, we prove that all right duo rings are right McCoy relative to any u.p.-monoid. We also show that for any nontrivial u.p.-monoid M, the class of right McCoy rings relative to M is contained in the class of right McCoy rings, and we present an example of a u.p.-monoid M for which this containment is strict.

Laurent Series Ring over Semiperfect Ring Can Not be Semiperfect

One of the main results of the article [2] says that, if a ring R is semiperfect and ϕ is an authomorphism of R, then the skew Laurent series ring R((x, ϕ)) is semiperfect. We will show that the above statement is not true. More precisely, we will show that, if the Laurent series ring R((x)) is semilocal, then R is semiperfect with nil Jacobson radical.

Lie solvability in matrix algebras

ABSTRACT If an algebra satisfies the polynomial identity (for short, is ), then is trivially Lie solvable of index (for short, is ). We prove that the converse holds for subalgebras of the upper triangular matrix algebra any commutative ring, and . We also prove that if a ring S is (respectively, ), then the subring of comprising the upper triangular matrices with constant main diagonal, is (respectively, ) for all . We also study two related questions, namely whether, for a field F, an subalgebra of for some n, with (F-)dimension larger than the maximum dimension of a subalgebra of , exists, and whether a subalgebra of with (the mentioned) maximum dimension, other than the typical subalgebras of with maximum dimension, which were described by Domokos and refined by van Wyk and Ziembowski, exists. Partial results with regard to these two questions are obtained.

Nilradicals of the unique product monoid rings

Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring R and a monoid M, we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring R[M]. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.