On rings whose Morita class is represented by matrix rings
A ring \(R\) is said to be an MM (short for ``Morita matrix'') ring if every ring which is Morita-equivalent to \(R\) is isomorphic to a full matrix ring over \(R\). Examples include rings for which all finitely-generated projective modules are free, but there are also many more interesting examples. In general an MM ring need not be indecomposable, but an MM ring which satisfies a polynomial identity (in particular, a commutative MM ring) is necessarily indecompossable. The paper is not restricted to commutative rings (for instance the final section is concerned with the class of von Neumann regular algebras known as ultramatricial algebras), but many of the results are for commutative rings and so for the remainder of this review we shall consider only the commutative case. A ring \(R\) is MM if and only if so also is the ring of formal power series over \(R\), but the MM property does not in general pass from \(R\) to the polynomial ring \(R[X]\) or to the partial quotient rings of \(R\). There is a lengthy discussion of the MM property for pullbacks, and in particular this gives an important method for constructing examples and counter-examples. If \(R\) is an MM ring then necessarily \(R\) is indecomposable and the Picard group \(\text{Pic}(R)\) is divisible, but these two conditions together are not sufficient for \(R\) to be MM. On the other hand, if \(R\) is Noetherian of Krull dimension at most one, then \(R\) is an MM ring if and only if \(R\) is indecomposable and \(\text{Pic}(R)\) is divisible. Thus a Dedekind domain is an MM ring if and only if its ideal class group is divisible.
scientific article; zbMATH DE number 1121701
scientific article