The notion of Morita-like equivalence has been introduced by Xu, Shum and T
urner-Smith in [13]. Their definition extends the usual Morita equivalence
far unital rings to a wider subclass of the class of all (non-unital) assoc
iative rings, namely the rat-rings. We study here this class of rings, show
ing its strong connections to other known classes of rings (s-unital rings,
local right identity rings, idempotent rings) and giving a complete descri
ption 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 ov
er a unital ring A.