Universal mapping properties of some pseudovaluation domains and related quasilocal domains

Ayache, Ahmed; Dobbs, David E.; Echi, Othman

doi:https://doi.org/10.1155/IJMMS/2006/72589

International Journal of Mathematics and Mathematical Sciences

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Volume 2006 | Article ID 072589 | https://doi.org/10.1155/IJMMS/2006/72589

Universal mapping properties of some pseudovaluation domains and related quasilocal domains

Ahmed Ayache,¹David E. Dobbs,²and Othman Echi³

Received24 Jan 2005

Revised05 Jan 2006

Accepted22 Jan 2006

Published18 May 2006

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.”

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Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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