Structure and topological conditions of NI rings

We first construct an NI ring but not 2-primal from given any 2-primal ring, in a simpler way than well-known examples. We study the structure of NI rings relating to strongly prime ideals and show that minimal strongly prime ideals can be lifted in NI rings. A ring is called (respectively weakly) pm if every (respectively strongly) prime ideal is contained in a unique maximal ideal in it. For a 2-primal ring R Sun proved that R is pm if and only if Max(R) is a retract of Spec(R) if and only if Spec(R) is normal. In the present note we prove for an NI ring R that R is weakly pm if and only if Max(R) is a retract of SSpec(R) if and only if SSpec(R) is normal, where SSpec(R) is the space of strongly prime ideals of R. We also prove that R is weakly pm if and only if R is pm when R is a symmetric ring. We lastly consider several kinds of extensions of NI rings.