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Bull. Korean Math. Soc. 2019; 56(2): 373-382

Online first article November 9, 2018      Printed March 1, 2019

https://doi.org/10.4134/BKMS.b180249

Copyright © The Korean Mathematical Society.

On strongly Gorenstein hereditary rings

Kui Hu, Hwankoo Kim, Fanggui Wang, Longyu Xu, Dechuan Zhou

Southwest University of Science and Technology; Hoseo University; Sichuan Normal University; Southwest University of Science and Technology; Southwest University of Science and Technology

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In this note, we mainly discuss strongly Gorenstein hereditary rings. We prove that for any ring, the class of $SG$-projective modules and the class of $G$-projective modules coincide if and only if the class of $SG$-projective modules is closed under extension. From this we get that a ring is an $SG$-hereditary ring if and only if every ideal is $G$-projective and the class of $SG$-projective modules is closed under extension. We also give some examples of domains whose ideals are $SG$-projective.

Keywords: strongly Gorenstein projective module, strongly Gorenstein hereditary ring, strongly Gorenstein Dedekind domain

MSC numbers: 13G05, 13D03

Supported by: This work is partially supported by National Natural Science Foundation of China(Grant No. 11671283 and Grant No.11401493), and the doctoral foundation of Southwest University of Science and Technology(No. 13zx7119).

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