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Mathematics
Algebra

Direct sums of injective and projective modules

Profile image of Pedro A Guil AsensioPedro A Guil Asensio

2010, Journal of Algebra

https://doi.org/10.1016/J.JALGEBRA.2010.05.020
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Abstract
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This paper extends a characterization of Σ-injective modules, originally established for injective modules, by providing new criteria based on the direct sums of injective and projective modules. Specific theorems are proven, highlighting that for an injective module M, various conditions regarding essential extensions and the structure of these modules are equivalent to Σ-injectivity. The implications of these results are also explored within the context of right noetherian rings, reinforcing the relationships between module properties.

FAQs

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What characterizes Σ-injective modules in terms of essential extensions?add

The paper demonstrates that a module M is Σ-injective if every essential extension of M(ℵ₀) is a direct sum of either injective or projective modules.

How does the injective hull relate to Σ-injectivity?add

The study reveals that the injective hull E(M) is Σ-injective if each essential extension of M(ℵ₀) is a direct sum of injective or projective modules.

What implications does Σ-injectivity have for right noetherian rings?add

It establishes that a ring R is right noetherian if each injective right R-module M's essential extensions are sums of injective or projective modules.

What proofs contribute to understanding Σ-injective modules?add

The authors adapt techniques from Guil Asensio and Simson to provide a more succinct proof of Theorem 2 concerning the characterization of Σ-injective modules.

Which module properties are crucial in the characterization of Σ-injective modules?add

The characterization relies on the absence of infinite direct sums within cyclic right R-modules related to quasi-injective modules, highlighting finiteness conditions.

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References (11)

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    We discuss various properties of a ring over which each simple module is Σ-injective. We shall consider associative rings with identity. Our modules will be unital right modules unless stated otherwise. The class of right V-rings was introduced by Villamayor [20]. A ring R is called a right V-ring if each simple right Rmodule is injective. It is a well-known unpublished result due to Kaplansky that a commutative ring is von Neumann regular if and only if it is a V-ring. The class of V-rings and, in particular, V-domains has been studied by many authors including Baccella ([1], [2]), Cozzens [5], Faith ([7], [8]), Fisher [10], Goodearl [14], Goursaud [12], Huynh et al [16], Kaplansky, Jain et al [17], Michler et al. [20], Osofsky [21], Page [22], Tyukavkin [24] and Tuganbaev [25]. Goursaud and Valette generalized the notion of V-ring and called a ring R to be a right Σ-V ring if each simple right R-module is Σ-injective [13]. Recall that a module M is called Σ-injective if every direct sum of copies of M is injective [6]. After defining these rings, Goursaud and Valette studied only the particular class of von Neumann regular right Σ-V rings [13]. Their paper [13] was published in 1975 but surprisingly in past 34 years no other paper except that of Baccella [1] even mentions Σ-V rings and this class of rings seems to have been forgotten. The present paper attempts to bring this interesting class of rings in light again for further research. Clearly, every right Σ-V ring is a right V-ring, however there are examples of right V-rings which are not right Σ-V rings.

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