Abstract
If(R,M)and(S,N)are quasilocal (commutative integral) domains andf:R→Sis a (unital) ring homomorphism, thenfis said to be astrong local homomorphism(resp.,radical local homomorphism) iff(M)=N(resp.,f(M)⊆Nand for eachx∈N, there exists a positive integertsuch thatxt∈f(M)). It is known that iff:R→Sis a strong local homomorphism whereRis a pseudovaluation domain that is not a field andSis a valuation domain that is not a field, thenffactors via a unique strong local homomorphism through the inclusion mapiRfromRto its canonically associated valuation overring(M:M). Analogues of this result are obtained which delete the conditions thatRandSare not fields, thus obtaining new characterizations of wheniRis 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.”
Highlights
Pseudovaluation domains were introduced by Hedstrom and Houston [21]
The morphisms that figure in these statements are the special type of local ring homomorphism f : (R,M) → (S,N) such that f (M) = N; we called such an f a “strong local homomorphism” in [3]
We begin by stating a result from [3] that is the culmination of an “orthogonality” study of pseudovaluation domains
Summary
Pseudovaluation domains (for short, PVDs) were introduced by Hedstrom and Houston [21]. (by [3, Proposition 3.9]), if u : R → T is a one-to-one slhomomorphism from a pseudo-᐀-domain R to a ᐀-domain T such that neither R nor T is a field, the following universal mapping property holds: for each ᐀-domain S which is not a field and each sl-homomorphism f : R → S, there exists a unique slhomomorphism f : T → S such that f u = f .
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