  We extend the classical notion of standardly stratified $k$-algebra (stated
for finite dimensional $k$-algebras) to the more general class of rings,
possibly without $1,$ with enough idempotents. We show that many of the
fundamental results, which are known for classical standardly stratified
algebras, can be generalized to this context. Furthermore, new classes of rings
appear as: ideally standardly stratified and ideally quasi-hereditary. In the
classical theory, it is known that quasi-hereditary and ideally
quasi-hereditary algebras are equivalent notions, but in our general setting
this is no longer true. To develop the theory, we use the well known connection
between rings with enough idempotents and skeletally small categories (ringoids
or rings with several objects).
