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- W2952014572 abstract "Let $(R, S,_Rnegthinspace M_S,_Snegthinspace N_R, f, g)$ be a general Morita context, and let $T=[{cc} R &_RM_S_SN_R & S]$ be the ring associated with this context. Similarly, let $T'=[{cc} R' & M' N' & S']$ be another Morita context ring. We study the set ${Iso}(T,T')$ of ring isomorphisms from $T$ to $T'$. Our interest in this problem is motivated by: (i) the problem to determine the automorphism group of the ring $T$, and (ii) the recovery of the non-diagonal tiles problem for this type of generalized matrix rings. We introduce two classes of isomorphisms from $T$ to $T'$, the disjoint union of which is denoted by ${Iso}_0(T,T')$. We describe ${Iso}_0(T,T')$ by using the $Z$-graded ring structure of $T$ and $T'$. Our main result characterizes ${Iso}_0(T,T')$ as the set consisting of all semigraded isomorphisms and all anti-semigraded isomorphisms from $T$ to $T'$, provided that the rings $R'$ and $S'$ are indecomposable and at least one of $M'$ and $N'$ is nonzero; in particular ${Iso}_0(T,T')$ contains all graded isomorphisms and all anti-graded isomorphisms from $T$ to $T'$. We also present a situation where ${Iso}_0(T,T')={Iso}(T,T')$. This is in the case where $R,S,R'$ and $S'$ are rings having only trivial idempotents and all the Morita maps are zero. In particular, this shows that the group of automorphisms of $T$ is completely determined." @default.
- W2952014572 created "2019-06-27" @default.
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- W2952014572 date "2011-06-30" @default.
- W2952014572 modified "2023-10-17" @default.
- W2952014572 title "Isomorphisms between Morita context rings" @default.
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- W2952014572 doi "https://doi.org/10.48550/arxiv.1106.6192" @default.
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