Gorenstein invariants under right Quasi-Frobenius extensions

Juxiang Sun; Guoqiang Zhao; Juxiang Sun; Guoqiang Zhao

doi:10.3934/era.2025158

Electronic Research Archive

2025, Volume 33, Issue 6: 3561-3570. doi: 10.3934/era.2025158

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Theory article

Gorenstein invariants under right Quasi-Frobenius extensions

Juxiang Sun ^{1
,
,},
Guoqiang Zhao ²

1.
School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu 476000, China
2.
School of Science, Hangzhou Dianzi University, Hangzhou 310018, China

Received: 24 February 2025 Revised: 01 June 2025 Accepted: 04 June 2025 Published: 10 June 2025

The focus of this work was to establish relations between Gorenstein projective modules linked by right quasi-Frobenius extensions of rings. As applications, the right quasi-Frobenius extensions of (weakly) Gorenstein algebras, Cohen-Macaulay finite algebras and the Cohen-Macaulay free algebra were studied. Suppose that $ \Gamma $ was a right quasi-Frobenius extension of an Artin algebra $ \Lambda $ with $ \Gamma $ a completely faithful left $ \Lambda $-module. We demonstrated that if $ \Gamma $ exhibited (weakly) Gorenstein properties, then $ \Lambda $ did so. Additionally, under the condition of $ \Gamma $ being a separable extension of $ \Lambda $, the converse became valid.
- right quasi-Frobenius extension,
- Gorenstein projective module,
- CM-finite algebra,
- CM-free algebra,
- Gorenstein algebra
Citation: Juxiang Sun, Guoqiang Zhao. Gorenstein invariants under right Quasi-Frobenius extensions[J]. Electronic Research Archive, 2025, 33(6): 3561-3570. doi: 10.3934/era.2025158

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Abstract

The focus of this work was to establish relations between Gorenstein projective modules linked by right quasi-Frobenius extensions of rings. As applications, the right quasi-Frobenius extensions of (weakly) Gorenstein algebras, Cohen-Macaulay finite algebras and the Cohen-Macaulay free algebra were studied. Suppose that $ \Gamma $ was a right quasi-Frobenius extension of an Artin algebra $ \Lambda $ with $ \Gamma $ a completely faithful left $ \Lambda $-module. We demonstrated that if $ \Gamma $ exhibited (weakly) Gorenstein properties, then $ \Lambda $ did so. Additionally, under the condition of $ \Gamma $ being a separable extension of $ \Lambda $, the converse became valid.

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