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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Acknowledgements
The authors would like to thank the anonymous referee for his/her careful reading and invaluable comments of the manuscript. Especially, we want to express our deep gratitude to his/her for commenting us the mistake that we made in the original proof of Theorem 3.10.
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Khojali, A., Zamani, N. V-Gorenstein injective modules preenvelopes and related dimension. Acta. Math. Sin.-English Ser. 33, 187–200 (2017). https://doi.org/10.1007/s10114-016-5559-3
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DOI: https://doi.org/10.1007/s10114-016-5559-3