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2012•Journal of Algebra and Its ApplicationsRequires access

ON RINGS OVER WHICH THE INJECTIVE HULL OF EACH CYCLIC MODULE IS Σ-EXTENDING

Sarapee Chairat, Dinh Van Huynh, Chitlada Somsup

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Abstract

Carl Faith (2003) introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings). Inspired by this we investigate rings over which every cyclic right R-module has a Σ-extending injective hull. We call such rings right CSE-rings and show that the class of right CSE-rings and that of right CSI-rings coincide. We also use other hulls of cyclic modules to define other classes of rings, and investigate their structure. We prove, among others, that a ring R is right QI if and only if the quasi-injective hull of each cyclic right module is Σ-injective.

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What this paper is about

Carl Faith (2003) introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings). Inspired by this we investigate rings over which every cyclic right R-module has a Σ-extending injective hull. We call such rings right CSE-rings and show that the class of right CSE-rings and that of right CSI-rings coincide. We also use other hulls of cyclic modules to define other classes of rings, and investigate their structure. We prove, among others, that a ring R is right QI if and only if the quasi-injective hull of each cyclic right module is Σ-injective.

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Available abstract

Carl Faith (2003) introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings). Inspired by this we investigate rings over which every cyclic right R-module has a Σ-extending injective hull. We call such rings right CSE-rings and show that the class of right CSE-rings and that of right CSI-rings coincide. We also use other hulls of cyclic modules to define other classes of rings, and investigate their structure. We prove, among others, that a ring R is right QI if and only if the quasi-injective hull of each cyclic right module is Σ-injective.

Key concepts: Mathematics, Injective function, Hull, Ring (chemistry), Class (philosophy), Injective module, Von Neumann regular ring, Combinatorics

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