Some variations of projectivity

Abstract

We prove that a ring [Formula: see text] has a module [Formula: see text] whose domain of projectivity consists of only some injective modules if and only if [Formula: see text] is a right noetherian right [Formula: see text]-ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor [Formula: see text]-modules is closed under direct summands if and only if [Formula: see text] is a right Bass ring. A ring [Formula: see text] is said to have no right max-p-middle class if every right [Formula: see text]-module is either projective or max-poor. It is shown that if a commutative noetherian ring [Formula: see text] has no right max-p-middle class, then [Formula: see text] is the ring direct sum of a semisimple ring [Formula: see text] and a ring [Formula: see text] which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field [Formula: see text] of [Formula: see text] has a proper [Formula: see text]-submodule which is not complete in its [Formula: see text]-topology. Then we show that a commutative noetherian hereditary ring [Formula: see text] has no right max-p-middle class if and only if [Formula: see text] is a semisimple ring.

Rings Whose Pure-Projective Modules Have Maximal or Minimal Projectivity Domain

“…Let R be an associative ring with identity throughout the article, and unless otherwise indicated, any module be a right R-module. Projectivity has been investigated from various angles in the recent studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The class {Y ∈ Mod-R : X is Y -projective} for a module X is referred to as the projectivity domain of X and is represented by Pr −1 (X) [16].…”

Lattice of Subinjective Portfolios of Modules

“…Recently, instead of simply categorizing modules as having a specific homological property, each module is allocated a relative domain that gauges the degree to which it possesses that particular property. In particular, several research papers have been devoted to the study of the injectivity, flatness, and projectivity level of modules [1][2][3][4][5][6][7][8][9][10][11].…”