Enlargements and Morita contexts for rings with involution

Abstract We study Morita equivalence for idempotent rings with involution. Following the ideas of Rieffel, we define Rieffel contexts, and we also introduce Morita $*$ -contexts and enlargements for rings with involution. We prove that two idempotent rings with involution have a joint enlargement if and only if they are connected by a unitary and full Rieffel context. These conditions are also equivalent to having a unitary and surjective Morita $*$ -context between those rings. We also examine how the mentioned conditions are connected to the existence of certain equivalence functors between the categories of firm modules over the given rings with involution.