International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 72589, 12 pages
doi:10.1155/IJMMS/2006/72589

Universal mapping properties of some pseudovaluation domains and related quasilocal domains

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

1Department of Mathematics, Faculty of Sciences, University of Bahrain, P.O. Box 32038, Isa Town, Bahrain
2Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA
3Department of Mathematics, Faculty of Sciences of Tunis, University Tunis-El Manar, Campus Universitaire, Tunis 2092, Tunisia

Received 24 January 2005; Revised 5 January 2006; Accepted 22 January 2006

Copyright © 2006 Ahmed Ayache et al. 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.

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.”