  We compute the associated prime ideals of the normalization modulo the ring,
and establish connections between different types of generalizations (resp.
specializations) of the normalization. This has some applications. For example,
we present a characterization of quasi-normality, in the format that was
conjectured by Vasconcelos. Also, we show in some cases, the normalization
modulo the ring, is homologically reduced. This provides a partial answer to
the conjecture of Matlis.
