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Some characterizations of regular modules.

Goro Azumaya (1990)

Publicacions Matemàtiques

Let M be a left module over a ring R. M is called a Zelmanowitz-regular module if for each x ∈ M there exists a homomorphism F: M → R such that f(x) = x. Let Q be a left R-module and h: Q → M a homomorphism. We call h locally split if for every x ∈ M there exists a homomorphism g: M → Q such that h(g(x)) = x. M is called locally projective if every epimorphism onto M is locally split. We prove that the following conditions are equivalent:(1) M is Zelmanowitz-regular.(2) every homomorphism into M...

Strong separativity over exchange rings

Huanyin Chen (2008)

Czechoslovak Mathematical Journal

An exchange ring R is strongly separative provided that for all finitely generated projective right R -modules A and B , A ⊕ A ≅ A ⊕ B ⇒ A ≅ B . We prove that an exchange ring R is strongly separative if and only if for any corner S of R , a S + b S = S implies that there exist u , v ∈ S such that a u = b v and S u + S v = S if and only if for any corner S of R , a S + b S = S implies that there exists a right invertible matrix a b * ∈ M 2 ( S ) . The dual assertions are also proved.

Strongly 2-nil-clean rings with involutions

Huanyin Chen, Marjan Sheibani Abdolyousefi (2019)

Czechoslovak Mathematical Journal

A * -ring R is strongly 2-nil- * -clean if every element in R is the sum of two projections and a nilpotent that commute. Fundamental properties of such * -rings are obtained. We prove that a * -ring R is strongly 2-nil- * -clean if and only if for all a ∈ R , a 2 ∈ R is strongly nil- * -clean, if and only if for any a ∈ R there exists a * -tripotent e ∈ R such that a - e ∈ R is nilpotent and e a = a e , if and only if R is a strongly * -clean SN ring, if and only if R is abelian, J ( R ) is nil and R / J ( R ) is * -tripotent. Furthermore, we explore the structure...

Structures naturelles des demi-groupes et des anneaux réguliers ou involutés

Jean Calmes (1994)

Mathématiques et Sciences Humaines

Certaines relations binaires sont définies sur les demi-groupes et les demi-groupes à involution. On examine comment elles peuvent en ordonner les éléments: notamment les idempotents, les éléments réguliers au sens de von Neumann, ceux qui possédent un inverse ponctuel ou de Moore-Penrose ; et en fonction aussi de conditions sur l'involution. Ces relations peuvent alors coïncider avec les ordres naturels des idempotents et des demi-groupes inverses, avec les ordres de Drazin et de Hartwig : elles...

The almost isomorphism relation for simple regular rings.

Pere Ara, Kenneth R. Goodearl (1992)

Publicacions Matemàtiques

A longstanding open problem in the theory of von Neumann regular rings is the question of whether every directly finite simple regular ring must be unit-regular. Recent work on this problem has been done by P. Menal, K. C. O'Meara, and the authors. To clarify some aspects of these new developments, we introduce and study the notion of almost isomorphism between finitely generated projective modules over a simple regular ring.

Über reguläre Z-Ringe

A. KERTÉSZ, O. STEINFELD (1974)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

Currently displaying 141 – 160 of 181