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:
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