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Invariant basis number of the ring of Morita context. (English) Zbl 0888.16003

A ring \(R\) is said to be an IBN ring if for every free left \(R\)-module \(F\) every two bases of \(F\) have the same cardinality. In this note conditions are shown when the ring of a Morita context \(T=\left(\begin{smallmatrix} R &M\\ N &S\end{smallmatrix}\right)\) is an IBN ring.
Theorem. Let \(M\) (\(N\)) be a finitely generated right (left) \(S\)-module. The ring \(T\) is an IBN ring if and only if \(R\) is an IBN ring or \(S\) is an IBN ring. Some corollaries are formulated, in particular: \(M_n(R)\) is an IBN ring if and only if \(R\) is an IBN ring.

MSC:

16D90 Module categories in associative algebras
16D40 Free, projective, and flat modules and ideals in associative algebras
16S50 Endomorphism rings; matrix rings
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References:

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