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41

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Introduction

Ryszard Mazurek currently works at the Faculty of Computer Science, Bialystok University of Technology, Poland. He does research in algebra.

**Skills and Expertise**

## Publications

Publications (41)

For any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S . We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a...

We characterize skew polynomial rings and skew power series rings that are reduced and right or left Archimedean.

We characterize skew polynomial rings and skew power series rings that are reduced and right or left Archimedean.

A skew generalized power series ring {R[[S,\omega,\leq]]} consists of all functions from a strictly ordered monoid {(S,\leq)} to a ring R whose support is artinian and narrow, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R . Special cases of this ring construction are skew...

We study which fields F can be represented as finite sums of proper subfields. We prove that for any \(n \ge 2\) every field F of infinite transcendence degree over its prime subfield can be represented as an unshortenable sum of n subfields, and every rational function field \(F = K(x_1, \ldots , x_n)\) can be represented as an unshortenable sum o...

ON ANNELIDAN, DISTRIBUTIVE, AND BÉZOUT RINGS - GREG MARKS, RYSZARD MAZUREK

A right chain ordered semigroup is an ordered semigroup whose right ideals form a chain. In this paper we study the ideal theory of right chain ordered semigroups in terms of prime ideals, completely prime ideals and prime segments, extending to these semigroups results on right chain semigroups proved in Ferrero et al. (J Algebra 292:574–584, 2005...

A skew generalized power series ring R[[S, ω]] consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R. Special cases of the skew...

We introduce the class of lineal rings, defined by the property that the lattice of right annihilators is linearly ordered. We obtain results on the structure of these rings, their ideals, and important radicals; for instance, we show that the lower and upper nilradicals of these rings coincide. We also obtain an affirmative answer to the Köthe Con...

We give a criterion for when idempotents of a ring R which commute modulo the Jacobson radical J(R) can be lifted to commuting idempotents of R. If such lifting is possible, we give extra information about the lifts. A "half-commuting" analogue is also proven, and this is used to give sufficient conditions for a ring to have the internal exchange p...

Given a positive integer n, a ring R is said to be n-semi-Armendariz if whenever fn = 0 for a polynomial f in one indeterminate over R, then the product (possibly with repetitions) of any n coefficients of f is equal to zero. A ring R is said to be semi-Armendariz if R is n-semi-Armendariz for every positive integer n. Semi-Armendariz rings are a g...

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.

Let R be a ring, S a strictly ordered monoid, and ω : S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, and (skew) monoid rings. We characterize when a skew generalized powe...

Given a semigroup S, we prove that if the upper nilradical is homogeneous whenever R is an S-graded ring, then the semigroup S must be cancelative and torsion-free. In case S is commutative the converse is true. Analogs of these results are established for other radicals and ideals. We also describe a large class of semigroups S with the property t...

Let R be a ring, S a strictly ordered monoid, and ω : S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) monoid rings, (skew) Mal'cev–Neumann series rings, and general...

For any commutative semigroup S and any positive integer m, the power function f:S→S defined by f(x)=x
m
is an endomorphism of S. In this paper we characterize finite cyclic semigroups as those finite commutative semigroups whose endomorphisms are power functions. We also prove that if S is a finite commutative semigroup with 1≠0, then every endomo...

In this paper we apply Ferrero-Sant'Ana's characterization of right distributive rings via saturations to prove that all right distributive rings are Armendariz relative to any unique product monoid. As an immediate consequence we obtain that all right distributive rings are Armendariz. We apply this result to give a new proof of the well-known fac...

A right-chain semigroup is a semigroup whose right ideals are totally ordered by set inclusion. The main result of this paper
says that if S is a right-chain semigroup admitting a ring structure, then either S is a null semigroup with two elements or sS=S for some s∈S. Using this we give an elementary proof of Oman’s characterization of semigroups...

We introduce a class of rings we call right Gaussian rings, defined by the property that for any two polynomials f, g over the ring R, the right ideal of R generated by the coefficients of the product fg coincides with the product of the right ideals generated by the coefficients of f and of g, respectively. Prüfer domains are precisely commutative...

Let R be a ring, S a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] 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 the case where S is positively ordered we g...

Let R be a ring, S a strictly ordered monoid, and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] 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. We study the (S,ω)-Armendariz condition o...

Let R be a ring, S a monoid and ω:S→End(R) a monoid homomorphism. In this paper we prove that if the monoid S is strictly totally ordered or S is commutative torsion-free cancellative semisubtotally ordered, then the ring R〚S,ω〛 of skew generalized power series with coefficients in R and exponents in S is a domain satisfying the ascending chain con...

We give necessary and sufficient conditions on a ring $R$ and an
endomorphism $\sigma$ of $R$ for the skew power series ring $R[[x;
\sigma]]$ to be right duo right Bézout. In particular, we prove
that $R[[x; \sigma]]$ is right duo right Bézout if and only if
$R[[x; \sigma]]$ is reduced right distributive if and only if
$R[[x; \sigma]]$ is right duo...

We introduce a class of ordered monoids defined by the existence of certain “unique products” with respect to artinian and
narrow subsets of the monoid. The logical relationships between this and other significant classes of monoids are explicated
with several examples. We conclude with results on skew generalized power series rings. The new class...

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 Ribenboim's 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...

Let R be a ring, S a strictly ordered monoid and ω:S→End(R) a monoid homomorphism. In this paper we obtain some necessary conditions for the skew generalized power series ring R〚S,ω〛 to be right (respectively left) uniserial, and we prove that these conditions are also sufficient when the monoid S is commutative or totally ordered.

In this article we study the problem when a left distributive algebra over a field determines an atom of the lattice of all radicals of rings. We show that this problem is strictly connected with the problem of describing extensions of left chain algebras and give a characterization of left chain algebras determining atoms.

Let R be a ring, and let S be a strictly ordered monoid. The generalized power series ring R[[S]] is a common generalization of polynomial rings, Laurent polynomial rings, power series rings, Laurent series rings, Mal'cev–Neumann series rings, monoid rings and group rings. In this paper, we examine which conditions on R and S are necessary and whic...

A semiprime segment of a ring R is a pair P2 P1 of semiprime ideals of R such that In P2 for all ideals I of R with P2 I P1. In this paper semiprime segments with P1 a comparizer ideal are classified as either simple, exceptional, or archimedean, extending to several classes of rings a classification known for right chain rings. These three types o...

Right chain semigroups are semigroups in which right ideals are linearly ordered by inclusion. Multiplicative semigroups of right chain rings, right cones, right invariant right holoids and right valuation semigroups are examples. The ideal theory of right chain semigroups is described in terms of prime and completely prime ideals, and a classifica...

A right ideal I of a ring R is said to be a comparizer right ideal of R if for any right ideals A, B of R, either A ⊆ B or BI ⊆ A. A ring R (with I) is called a right pseudo-chain ring if for any nonunit a ∈ R, aR is a comparizer right ideal of R. These rings are generalizations of right chain rings and commutative pseudo-valuation rings. In this p...

Let T be a right chain ring with non-zero maximal ideal J. We study subrings R of T containing J and determine conditions for R to be a right distributive (right Bezout) ring. As a consequence we obtain a structure theorem for semiprime right distributive (right Bezout) rings with non-zero left waists.

A right ideal I of a ring R is called a comparizer right ideal if for all right ideals A, B of R, either A ⊆ B or BI ⊆ A. For every ring R there exists the largest comparizer ideal C1(R) of R, and higher comparizer ideals Cα(R) can be defined inductively. In this paper, comparizer right ideals and relationships between the iterated comparizer ideal...

Let be a nonzero ordinal such that + = for every ordinal

Let λ be a property that a lattice of submodules of a module may possess and which is preserved under taking sublattices and isomorphic images of such lattices and is satisfied by the lattice of subgroups of the group of integer numbers. For a ring R the lower radical Λ generated by the class λ(R) of R-modules whose lattice of submodules possesses...

This chapter discusses distributive radical. Many authors study rings with distributive lattice of left ideals. The restrictive assumption of distributivity forces positive answers to some questions of ring theory. It was observed that this remains true for a more general class of rings, namely rings that are sums of distributive left ideals. Becau...

Monoid S nazywamy u.p.-monoidem (unique product monoid), jeżeli dla dowol-nych niepustych i skończonych podzbiorów A, B zbioru S istnieje element s ∈ AB, który tylko na jeden sposób może być zapisany w postaci s = ab, gdzie a ∈ A i b ∈ B. Takie monoidy grają ważną rolę w badaniach pierścieni grupowych i półgrupowych ([2], [5]). Celem referatu jest...