Rings and modules characterized by opposites of injectivity. (English) Zbl 1315.16011
A module \(M\) is said to be \(N\)-subinjective if every homomorphism from \(N\) to \(M\) can be extended to a homomorphism from \(E(N)\) to \(M\). The authors of this paper call a module \(M\) to be a test for injectivity by subinjectivity (t.i.b.s., for short) if the only modules which are \(M\)-subinjective are the injective ones. It is shown that a t.i.b.s. module exists over every ring. This paper studies structure of rings over which each module is injective or a t.i.b.s. It is also shown that a ring \(R\) is right hereditary and right Noetherian if and only if \(R\) as a right \(R\)-module is a t.i.b.s.
Reviewer: Ashish K. Srivastava (Saint Louis)
MSC:
| 16D80 | Other classes of modules and ideals in associative algebras |
| 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) |
Keywords:
injective modules; subinjective modules; injective hulls; injectivity conditions; direct products; Artinian serial rings; tests for injectivity by subinjectivity; t.i.b.s. modulesReferences:
| [1] | Alahmadi, A. N.; Alkan, M.; López-Permouth, S. R., Poor modules: The opposite of injectivity, Glasg. Math. J., 52A, 7-17 (2010) · Zbl 1228.16004 |
| [2] | Alizade, R.; Bilhan, G.; Smith, P. F., Modules whose maximal submodules have supplements, Comm. Algebra, 29, 6, 2389-2405 (2001) · Zbl 0989.16001 |
| [3] | Aydoǧdu, P.; López-Permouth, S. R., An alternative perspective on injectivity of modules, J. Algebra, 338, 207-219 (2011) · Zbl 1246.16005 |
| [4] | Baba, Y.; Oshiro, K., Classical Artinian Rings and Related Topics (2009), World Scientific: World Scientific Singapore · Zbl 1204.16001 |
| [5] | Damiano, R. F., A right PCI-ring is right Noetherian, Proc. Amer. Math. Soc., 77, 1, 11-14 (1979) · Zbl 0425.16022 |
| [6] | Dung, N. V.; Huynh, D. V.; Smith, P. F.; Wisbauer, R., Extending Modules, Pitman Res. Notes Math. Ser., vol. 313 (1994) · Zbl 0841.16001 |
| [7] | Er, N.; López-Permouth, S.; Sökmez, N., Rings whose modules have maximal or minimal injectivity domains, J. Algebra, 330, 404-417 (2011) · Zbl 1227.16004 |
| [8] | Faith, C., Rings and Things and a Finite Array of Twentieth Century Associative Algebra, Math. Surveys Monogr., vol. 65 (2004), Amer. Math. Soc. |
| [9] | Faith, C., When are proper cyclics injective?, Pacific J. Math., 45, 1, 97-112 (1973) · Zbl 0258.16024 |
| [10] | Goodearl, K., Singular Torsion and the Splitting Properties, Mem. Amer. Math. Soc., vol. 124 (1972) · Zbl 0242.16018 |
| [11] | Lam, T. Y., Lectures on Modules and Rings (1999), Springer: Springer Berlin, Heidelberg, New York · Zbl 0911.16001 |
| [12] | Mohamed, S. H.; Müller, B. J., Continuous and Discrete Modules, London Math. Soc. Lecture Note Ser., vol. 147 (1990), Cambridge University Press · Zbl 0701.16001 |
| [13] | Osofsky, B., Rings all of whose finitely generated modules are injective, Pacific J. Math., 14, 645-650 (1964) · Zbl 0145.26601 |
| [14] | Trlifaj, J., Whitehead test modules, Trans. Amer. Math. Soc., 348, 4, 1521-1554 (1996) · Zbl 0865.16006 |
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