Rings having a Morita-like equivalence

The notion of Morita-like equivalence has been introduced by Xu, Shum and Turner-Smith in [13]. Their definition extends the usual Morita equivalence far unital rings to a wider subclass of the class of all (non-unital) associative rings, namely the rat-rings. We study here this class of rings, showing its strong connections to other known classes of rings (s-unital rings, local right identity rings, idempotent rings) and giving a complete description of xst-rings in terms of these known classes. We also show that right xst-rings which are Morita-like equivalent to unital rings come from dense subrings of the endomorphism ring End(M-A) of some generator module M-A over a unital ring A.