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Strongly (\(\mathscr {X},\mathscr {Y},\mathscr {Z}\))-Gorenstein Modules and Applications

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Abstract

In this paper, we introduce strongly (\(\mathscr {X},\mathscr {Y},\mathscr {Z}\))-Gorenstein modules, where \(\mathscr {X},\mathscr {Y},\mathscr {Z}\) are additive full subcategories of R-\(\mathrm Mod\). These modules provide a new characterization of \(\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})\)-modules, that every \(\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})\)-module is a direct summand of a certain strongly (\(\mathscr {X},\mathscr {Y},\mathscr {Z}\))-Gorenstein module. Another application in global dimensions is given: the global left \({\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})}\)-dimension is equal to the global right \({\mathcal {G}(\mathscr {\mathscr {X^{'}},\mathscr {Y^{'}},\mathscr {Z^{'}}})}\)-dimension over any ring R, where \(\mathscr {X^{'}},\mathscr {Y^{'}},\mathscr {Z^{'}}\) are additive full subcategories of R-\(\mathrm{Mod}\) related to \(\mathscr {X},\mathscr {Y},\mathscr {Z}\).

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Acknowledgements

I would like to thank the anonymous referee for constructive comments and useful suggestions. A part of this work was written, while the author was visiting The Ohio State University. During this period, I have to thank Professor Yousif for warm-hearted reception, for the careful guidance and for the continuous encouragement. At the same time, I would like to express my gratitude to the China Scholarship Council (CSC) and the National Natural Science Foundation of China (11401339) that provide me with the financial support.

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  1. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Liuzhen Kong

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  1. Liuzhen Kong
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Correspondence to Liuzhen Kong.

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Communicated by Siamak Yassemi.

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Kong, L. Strongly (\(\mathscr {X},\mathscr {Y},\mathscr {Z}\))-Gorenstein Modules and Applications. Bull. Iran. Math. Soc. 46, 503–517 (2020). https://doi.org/10.1007/s41980-019-00272-w

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  • DOI: https://doi.org/10.1007/s41980-019-00272-w

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