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Rings whose modules have maximal submodules.

Carl Faith (1995)

Publicacions Matemàtiques

A ring R is a right max ring if every right module M ≠ 0 has at least one maximal submodule. It suffices to check for maximal submodules of a single module and its submodules in order to test for a max ring; namely, any cogenerating module E of mod-R; also it suffices to check the submodules of the injective hull E(V) of each simple module V (Theorem 1). Another test is transfinite nilpotence of the radical of E in the sense that radα E = 0; equivalently, there is an ordinal α such that radα(E(V))...

Rings whose nonsingular right modules are R -projective

Yusuf Alagöz, Sinem Benli, Engin Büyükaşık (2021)

Commentationes Mathematicae Universitatis Carolinae

A right R -module M is called R -projective provided that it is projective relative to the right R -module R R . This paper deals with the rings whose all nonsingular right modules are R -projective. For a right nonsingular ring R , we prove that R R is of finite Goldie rank and all nonsingular right R -modules are R -projective if and only if R is right finitely Σ - C S and flat right R -modules are R -projective. Then, R -projectivity of the class of nonsingular injective right modules is also considered. Over right...

Rings whose proper factors are right perfect

Alberto Facchini, Catia Parolin (2011)

Colloquium Mathematicae

We show that practically all the properties of almost perfect rings, proved by Bazzoni and Salce [Colloq. Math. 95 (2003)] for commutative rings, also hold in the non-commutative setting.

Rings with zero intersection property on annihilators: Zip rings.

Carl Faith (1989)

Publicacions Matemàtiques

Zelmanowitz [12] introduced the concept of ring, which we call right zip rings, with the defining properties below, which are equivalent:(ZIP 1) If the right anihilator X⊥ of a subset X of R is zero, then X1⊥ = 0 for a finite subset X1 ⊆ X.(ZIP 2) If L is a left ideal and if L⊥ = 0, then L1⊥ = 0 for a finitely generated left ideal L1 ⊆ L.In [12], Zelmanowitz noted that any ring R satisfying the d.c.c. on anihilator right ideals (= dcc ⊥) is a right zip ring, and hence, so is any subring of R. He...

Roots of Nakayama and Auslander-Reiten translations

Helmut Lenzing, Andrzej Skowroński (2000)

Colloquium Mathematicae

We discuss the roots of the Nakayama and Auslander-Reiten translations in the derived category of coherent sheaves over a weighted projective line. As an application we derive some new results on the structure of selfinjective algebras of canonical type.

Rota-Baxter operators and Bernoulli polynomials

Vsevolod Gubarev (2021)

Communications in Mathematics

We develop the connection between Rota-Baxter operators arisen from algebra and mathematical physics and Bernoulli polynomials. We state that a trivial property of Rota-Baxter operators implies the symmetry of the power sum polynomials and Bernoulli polynomials. We show how Rota-Baxter operators equalities rewritten in terms of Bernoulli polynomials generate identities for the latter.

Rough semigroups and rough fuzzy semigroups based on fuzzy ideals

Qiumei Wang, Jianming Zhan (2016)

Open Mathematics

In this paper, we firstly introduce a special congruence relation U(μ, t) induced by a fuzzy ideal μ in a semigroup S. Then we define the lower and upper approximations based on a fuzzy ideal in semigroups. We can establish rough semigroups, rough ideals, rough prime ideals, rough fuzzy semigroups, rough fuzzy ideals and rough fuzzy prime ideals according to the definitions of rough sets and rough fuzzy sets. Furthermore, we shall consider the relationships among semigroups and rough semigroups,...

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