Self extending rings

A ring R is an IPQ (isomorphic proper quotient)-ring if R ≃ R/A for every proper ideal A ◁ R. If every ideal A ⊴ R satisfies: either R ≃ A or R ≃ R/A, then R is called an SE (self extending)-ring. It is shown that with one exception, an abelian group G is the additive group of an IPQ-ring if and only if G is the additive group of an SE-ring. The one exception is the infinite cyclic group Z. The zeroring with additive group Z is an SE-ring, but a ring with infinite cyclic additive group is not an IPQ-ring. Since the structure of the additive groups of IPQ-rings is known, the structure of the additive groups of SE-rings is completely determined.