Algebras with radical square zero are either self-injective or CM-free
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Abstract:
An artin algebra is called CM-free provided that all its finitely generated Gorenstein projective modules are projective. We show that a connected artin algebra with radical square zero is either self-injective or CM-free. As a consequence, we prove that a connected artin algebra with radical square zero is Gorenstein if and only if its valued quiver is either an oriented cycle with the trivial valuation or does not contain oriented cycles.References
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Additional Information
- Xiao-Wu Chen
- Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Chinese Academy of Sciences, Hefei 230026, Anhui, People’s Republic of China
- Email: xwchen@mail.ustc.edu.cn
- Received by editor(s): June 10, 2010
- Received by editor(s) in revised form: November 9, 2010
- Published electronically: May 16, 2011
- Additional Notes: The author is supported by the Special Foundation of the President of the Chinese Academy of Sciences (No. 1731112304061) and by the National Natural Science Foundation of China (No. 10971206).
- Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 93-98
- MSC (2010): Primary 18G25, 16G10, 16G50
- DOI: https://doi.org/10.1090/S0002-9939-2011-10921-3
- MathSciNet review: 2833520