Perhaps what you want is something along the lines of Rieffel's strong Morita equivalence of $C^\ast$-algebras (see http://math.berkeley.edu/~rieffel/papers/morita_equivalence.pdfsee, for instance, http://math.berkeley.edu/~alanw/242papers99/bursztyn, and pretty much any introductory account of operator-algebraic noncommutative geometry), except forgetting all the functional-analytic nuances?
Perhaps what you want is something along the lines of Rieffel's strong Morita equivalence of $C^\ast$-algebras (see http://math.berkeley.edu/~rieffel/papers/morita_equivalence.pdf and pretty much any introductory account of operator-algebraic noncommutative geometry), except forgetting all the functional-analytic nuances?