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A Generalization of Baer's Lemma

Molly Dunkum (2009)

Czechoslovak Mathematical Journal

There is a classical result known as Baer’s Lemma that states that an R -module E is injective if it is injective for R . This means that if a map from a submodule of R , that is, from a left ideal L of R to E can always be extended to R , then a map to E from a submodule A of any R -module B can be extended to B ; in other words, E is injective. In this paper, we generalize this result to the category q ω consisting of the representations of an infinite line quiver. This generalization of Baer’s Lemma...

Kaplansky classes

Edgar E. Enochs, J. A. López-Ramos (2002)

Rendiconti del Seminario Matematico della Università di Padova

Relative multiplication and distributive modules

José Escoriza, Blas Torrecillas (1997)

Commentationes Mathematicae Universitatis Carolinae

We study the construction of new multiplication modules relative to a torsion theory τ . As a consequence, τ -finitely generated modules over a Dedekind domain are completely determined. We relate the relative multiplication modules to the distributive ones.

The torsion theory and the Melkersson condition

Takeshi Yoshizawa (2020)

Czechoslovak Mathematical Journal

We consider a generalization of the notion of torsion theory, which is associated with a Serre subcategory over a commutative Noetherian ring. In 2008 Aghapournahr and Melkersson investigated the question of when local cohomology modules belong to a Serre subcategory of the module category. In their study, the notion of Melkersson condition was defined as a suitable condition in local cohomology theory. One of our purposes in this paper is to show how naturally the concept of Melkersson condition...

z⁰-Ideals and some special commutative rings

Karim Samei (2006)

Fundamenta Mathematicae

In a commutative ring R, an ideal I consisting entirely of zero divisors is called a torsion ideal, and an ideal is called a z⁰-ideal if I is torsion and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We prove that in large classes of rings, say R, the following results hold: every z-ideal is a z⁰-ideal if and only if every element of R is either a zero divisor or a unit, if and only if every maximal ideal in R (in general, every prime z-ideal) is a z⁰-ideal,...

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