When Cyclic Modules Have Σ-Injective Hulls

Abstract

A theorem of Cartan-Eilenberg (Cartan, H., Eilenberg, S. (1956). Homological Algebra. Princeton: Princeton University Press, pp. 390.) states that a ring Ris right Noetherian iff every injective right module is Σ-incentive. The purpose of this paper is to study rings with the property, called right CSI, that, all cyclic right R-modules have Σ-injective hulls, i.e., injective hulls of cyclic right R-modules are Σ-injective. In this case, all finitely generated right R-modules have Σ-injective hulls, and this implies that Ris right Noetherian for a lengthy list of rings, most notably, for Rcommutative, or when Rhas at most finitely many simple right R-modules, e.g., when Ris semilocal. Whether all right CSIrings are Noetherian is an open question. However, if in addition, R/rad Ris either right Kasch or von Neuman regular (=VNR), or if all countably generated (sermisimple) right R-modules have Σ-injective hulls then the answer is affirmative. (See Theorem A.) We also prove the dual theorems for Δ-injective modules.

Dedicated to Ahmad Shamsuddin (1951–2001), in memoriam.

Key Words:

• Σ-injective
• Noetherian
• Δ-injective
• Artinian
• Semilocal
• Von Neumann regular
• Kasch

1991 Mathematics Subject Classification:

• Primary: 16D50, 16P40, 16P60
• Secondary: 16E50, 16L30, 13C11

Acknowledgment

The author thanks the referee for his thorough reading of the manuscript and for suggestions which greatly improved its readability and other desiderata.

Notes

Dedicated to Ahmad Shamsuddin (1951–2001), in memoriam.