Invariant basis number of the ring of Morita context (Q1368195)

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.