1 research outputs found
Universal mapping properties of some pseudovaluation domains and related quasilocal domains
If (R,M) and (S,N)
are quasilocal (commutative integral)
domains and f:RâS 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)âN
and for each xâN, there exists a positive
integer t
such that xtâf(M)). It is known that if
f:RâS
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 âpseudovaluation domain that is not a
fieldâ condition is replaced by the APVDs of Badawi-Houston and the âstrong local homomorphismâ conditions are replaced by âradical local homomorphism.