/pdf/rings-over-which-certain-modules-are-injective-i4gg9pdwd4.pdf

Rings over which certain modules are injective.

Rings whose modules have maximal or minimal injectivity domains

Am oduleM is ⊕-supplemented if every submodule of M has a supplement which is a direct summand of M. In this paper, we show that a quotient of a ⊕-supplemented module is not in general ⊕-supplemented. We prove that over a commutative ring R, every finitely generated ⊕-supplemented R-module M having dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ring R is semisimple if and only if the class of ⊕-supplemented R-modules coincides with the class of injective R-modules. The structure of ⊕-supplemented modules over a commutative principal ideal ring is completely determined.

/pdf/on-some-properties-of-supplemented-modules-uljl7330j2.pdf

On some properties of ⊕-supplemented modules

Relative injectivity and chain conditions

/pdf/krull-dimension-and-noetherianness-2e4qxkjbu0.pdf

Krull dimension and noetherianness

A ring R is called a V-ring, respectively SSI-ring, respectively QII-ring if simple, respectively semisimple, respectively quasi-injective, right R-modules are injective. We show that R is SSI if and only if R is a right noetherian V-ring and that any SSI-ring is a finite ring direct sum of simple SSI-rings. We show that if R is left noetherian and SSI then R is QII provided R is hereditary and that in order for R to be hereditary it suffices that maximal right ideals of R be reflexive. An example of Cozzens is cited to show these rings need not be artinian. The rings R which have the property that quasi-injective right Rmodules are injective (herein called QII-rings) have been little mentioned in the literature though related classes of rings have been extensively examined. Consider the classes of R-modules consisting of (1) all Rmodules, (2) injective R-modules, (3) projective R-modules, (4) quasiinjective R-modules, and (5) quasi-projective R-modules. Of all the nontrivial inclusions one can assume between two of these classes, all but one has been shown to characterize the ring R as either semisimple artinian, quasi-Frobenius, or uniserial. That one excepted is the class of QII-rings whose internal structure, it seems, has not been completely determined. This paper does not attempt a general structure theorem but investigates the connection between QII-rings and rings whose semisimple right modules are injective (SSI-rings). Clearly every QII-ring is an SSI-ring. We show that SSI-rings are the same as right noetherian, V-rings and that an SSI-ring is a finite ring direct sum of simple SSI-rings. Finally, it is shown that a left noetherian, right SSI-ring which is hereditary is a QII-ring and that for such a ring to be hereditary it suffices that maximal right ideals be reflexive. All rings are assumed to be associative rings with identity and all modules are unital, right modules unless otherwise indicated. V, SSI, and QII refer to the appropriate condition on the right modules of the ring. Received by the editors July 6, 1971. AMS 1970 subject class.ifcations. Primary 16A52, 16A62; Secondary 16A46.

https://www.ams.org/proc/1972-033-02/S0002-9939-1972-0310009-7/S0002-9939-1972-0310009-7.pdf

Rings whose quasi-injective modules are injective

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Rings whose quasi-injective modules are injective