Universal mapping properties of some pseudovaluation domains and related quasilocal domains

Ahmed Ayache,David E Dobbs,Othman Echi

doi:10.1155/ijmms/2006/72589

Abstract

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 xt ∈ 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 iR 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 iR 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.”

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

Introduction

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 .

Objectives

Results

Conclusion