A theorem of Cartan-Eilenberg (Cartan, H., Eilenberg, S. (1956). Homological Algebra. Princeton: Princeton University Press, pp. 390.) states that a ring R is right Noetherian iff every injective right module is Sigma-incentive. The purpose of this paper is to study rings with the property, called right CSI, that, all cyclic right R-modules have Sigma-injective hulls, i.e., injective hulls of cyclic right R-modules are Sigma-injective. In this case, all finitely generated right R-modules have Sigma-injective hulls, and this implies that R is right Noetherian for a lengthy list of rings, most notably, for R commutative, or when R has at most finitely many simple right R-modules, e.g., when R is semilocal. Whether all right CSI rings are Noetherian is an open question. However, if in addition, R/rad R is either right Kasch or von Neuman regular (= VNR), or if all countably generated (sermisimple) right R-modules have Sigma-injective hulls then the answer is affirmative. (See Theorem A.) We also prove the dual theorems for Delta-injective modules. [References: 28]
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