<Emphasis Type="Italic">V</Emphasis>-Gorenstein injective modules preenvelopes and related dimension |
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Authors: | Ahmad Khojali Naser Zamani |
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Institution: | Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil, P. O. Box 56199-11367, Iran |
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Abstract: | Let R and S be associative rings and S V R a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom R (I V (R),?) and Hom R (?,I V (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 i and I i , i ∈ N0, such that N ? Im(I 0 → I 0). 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 V (R) which leads to the fact that V-Gorenstein injective modules admit exact right I V (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 V (R)-injective dimension. |
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Keywords: | V-Gorenstein injective module V-injective module semidualizing module Auslander class |
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