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| Stable equivalences of Morita type is a basic equivalent relation between finite dimensional algebras, introduced by Brou'{e} in the study of modular representation theory of finite groups. It is closely related to the famous Abelian Defect Group Conjecture. Later, Liu and Xi give a series of methods to construct stable equivalences of Morita type between general finite dimensional algebras. In particular, they proved that: If a stable equivalence of Morita type between two basic self-injective algebras $A$ and $B$ takes the socle $soc(P)$ of an indecomposable projective-injective $A$-module $P$ to the socle $soc(P')$ of an indecomposable projective-injective $B$-module $P'$, then the quotient algebras $A/soc(A)$ and $B/soc(B)$ are stably equivalent of Morita type. In this note, generalize this result by given a more general construction of stable equivalences of Morita type. Particularly, we improve the above mentioned result by remove the condition that $A$ and $B$ are self-injective. |
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| Keywords:finite dimensional algebra, quotient algebra, stable equivalence of Morita type |
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