Universal mapping properties of some pseudovaluation domains and related quasilocal domains

If (R,M) and (S,N) are quasilocal (commutative integral) domains and f : R → S is a (unital) ring homomorphism, then f is said to be a strong local homomorphism (resp., radical local homomorphism) if f(M) = N (resp., f(M) ⊆ N and for each x ∈ N, there exists a positive integer t such that x^t ∈ f(M)). It is known that if f : R → S is a strong local homomorphism where R is a pseudovaluation domain that is not a field and S is a valuation domain that is not a field, then f factors via a unique strong local homomorphism through the inclusion map i_R from R to its canonically associated valuation overring (M : M). Analogues of this result are obtained which delete the conditions that R and S are not fields, thus obtaining new characterizations of when i_R is integral or radicial. Further analogues are obtained in which the "pseudovaluation domain that is not a field" condition is replaced by the APVDs of Badawi-Houston and the "strong local homomorphism" conditions are replaced by "radical local homomorphism.".