Let S be the class of all rings R which have 1 and satisfy this condition:
for every "non-zero" right ideal I of R there exists a "proper" right ideal J of R such that I + J = R. (The + here is not necessarily direct.)
All semisimple rings are in S and (commutative) local rings which are not fields are not in S. The ring of integers Z is also in S and so S properly contains the class of semisimple rings.
Will this condition by itself force an element of S to have any (known, interesting) structure?
A more important question:
What about simple rings which are in S? For example, do they have to be semisimple? (Unlikely!)