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- Title
V-Gorenstein injective modules preenvelopes and related dimension.
- Authors
Khojali, Ahmad; Zamani, Naser
- Abstract
Let R and S be associative rings and V a semidualizing ( S- R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom( I ( R),−) and Hom(−, I ( R)) exact exact complex $$ \cdots \to {I_1}\xrightarrow{{{d_0}}}{I_0} \to {I^0}\xrightarrow{{{d_0}}}{I^1} \to \cdots $$ of V-injective modules I and I , i ∈ N, such that N ≅ Im( I → I ). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class A ( R) which leads to the fact that V-Gorenstein injective modules admit exact right I ( R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V-Gorenstein injective if and only if N ⊕ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if $$Ext_{{I_V}\left( R \right)}^{ \geqslant n + 1}\left( {I,N} \right) = 0$$ for all modules I with finite I ( R)-injective dimension.
- Subjects
HOMOMORPHISMS; GORENSTEIN rings; FREE resolutions (Algebra); INTEGERS; LINEAR operators
- Publication
Acta Mathematica Sinica, 2017, Vol 33, Issue 2, p187
- ISSN
1439-8516
- Publication type
Article
- DOI
10.1007/s10114-016-5559-3