On rings whose Morita class is represented by matrix rings Original Research Article

We define and study Morita Matrix (MM) rings, i.e. rings whose only Morita equivalent rings are (up to isomorphism) the matrix rings over them. In the description of rings of this type, Picard progenerators and (in the commutative case) the Picard group play a significant role. For a wide class of rings (including commutative ones) indecomposability is necessary to be an MM ring. For commutative rings an additional necessary condition is that the Picard group be divisible. In a number of special cases these two conditions will also become sufficient. The behaviour of MM rings under some of the usual constructions (i.e. rings of polynomials and power series, pullbacks, etc.) is also examined and from this numerous examples of MM rings are derived. Further non-commutative examples are found amongst ultramatricial von Neumann regular rings.