What question did this study set out to answer?

The aim is to introduce and analyze projective modules and their global dimensions in the context of Noetherian rings.

February 28, 2026Journal of Algebra and Its Applications0 citations

On τq-projectivity and τq-simplicity

Key Points

The aim is to introduce and analyze projective modules and their global dimensions in the context of Noetherian rings.
Introduced the concept of projective modules using strongly Lucas modules.
Investigated the global dimension of rings, specifically for Noetherian rings.
Studied the conditions under which all modules are projective and identified semisimple rings.
Demonstrated that for a Noetherian ring, specific global dimension properties hold true.
Established that a ring is semisimple if particular conditions regarding its structure are satisfied.
Showed that various types of injective modules can be defined based on the properties of the rings studied.

Abstract

In this paper, we first introduce and study the notion of Formula: see text-projective modules via strongly Lucas modules, and then investigate the Formula: see text-global dimension Formula: see text-gl.dimFormula: see text of a ring Formula: see text. We obtain that if Formula: see text is a Formula: see text-Noetherian ring, then Formula: see text-gl.dimFormula: see text-gl.dimFormula: see textgl.dimFormula: see text. Finally, we study the rings over which all modules are Formula: see text-projective (i.e., Formula: see text-semisimple rings). In particular, we show that a ring Formula: see text is a Formula: see text-semisimple ring if and only if Formula: see text (or Formula: see text, or Formula: see text) is a semisimple ring, if and only if Formula: see text is a reduced ring with Formula: see text finite, if and only if every reg-injective (or semireg-injective, or Lucas, or strongly Lucas) module is injective.

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