×

Modules which are invariant under monomorphisms of their injective hulls. (English) Zbl 1104.16003

From the authors’ abstract: In this paper, certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring \(R\) is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of \(R\) to \(R^{(\mathbb{N})}\) can be extended to \(R\). Also known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is quasi-injective. The following results are obtained: \(M\) is quasi-injective if and only if \(M\) is pseudo-injective and \(M^2\) is CS. Furthermore, if \(M\) is a direct sum of uniform modules, then \(M\) is quasi-injective if and only if \(M\) is pseudo-injective. As a consequence of this it is shown that over a right Noetherian ring \(R\), quasi-injective modules are precisely pseudo-injective CS modules.
Comments: With the “key lemma” 3.5 the authors improve a result of the reviewer (Theorem 3.4 in the reviewer’s paper [Commun. Algebra 33, No. 2, 361-369 (2005; Zbl 1077.16004)]) by removing the nonsingularity of the module \(M\). Furthermore, with Theorems 3.4 and 3.6, the authors give partial answers to a question raised by the reviewer (Question 3.5 in the reviewer’s same paper, and also mentioned in the above abstract).

MSC:

16D50 Injective modules, self-injective associative rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16W20 Automorphisms and endomorphisms

Citations:

Zbl 1077.16004
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dung, Extending modules 313 (1994)
[2] DOI: 10.1216/rmjm/1181071648 · Zbl 0942.16008
[3] Alamelu, J. Indian Math. Soc. 39 pp 121– (1975)
[4] DOI: 10.1081/AGB-100002397 · Zbl 0983.16001
[5] Tuganbaev, Trudy Sem. Petrovsk. 4 pp 241– (1978)
[6] DOI: 10.2307/2040636 · Zbl 0303.16013
[7] Singh, Riv. Mat. Univ. Parma 9 pp 59– (1968)
[8] Osofsky, Pacific J. Math. 14 pp 646– (1964) · Zbl 0145.26601
[9] Okado, Math. Japonica 29 pp 939– (1984)
[10] Jain, Canad. Math. Bull. 18 pp 359– (1975) · Zbl 0326.16023
[11] Mohamed, continuous and Discrete Modules (1990)
[12] Jain, J. Math. Sci. 2 pp 23– (1967)
[13] DOI: 10.2307/2154777 · Zbl 0852.16013
[14] Goodearl, Singular torsion and the splitting properties (1972) · Zbl 0242.16018
[15] Er, East-West J. Math. 4 pp 1– (2002)
[16] DOI: 10.1081/AGB-200040917 · Zbl 1077.16004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.