The trailing terms ideal over a valuation domain

2021 ◽  
pp. 1-6
Author(s):  
Ihsen Yengui
Keyword(s):  
Valuation Domain

scholarly journals Universal mapping properties of some pseudovaluation domains and related quasilocal domains

2006 ◽  
Vol 2006 ◽  
pp. 1-12
Author(s):  
Ahmed Ayache ◽  
David E. Dobbs ◽  
Othman Echi
Keyword(s):  
Positive Integer ◽  
Field Condition ◽  
Ring Homomorphism ◽  
Valuation Domain ◽  
Integral Domains ◽  
Unital Ring ◽  
Local Homomorphism

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.”

Grobner bases over a dual valuation domain

2013 ◽  
Vol 7 ◽  
pp. 539-548
Author(s):  
Andre Saint Eudes Mialebama Bouesso
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases ◽  
Valuation Domain

NORMAL PAIRS WITH ZERO-DIVISORS

2011 ◽  
Vol 10 (02) ◽  
pp. 335-356 ◽  
Author(s):  
DAVID E. DOBBS ◽  
JAY SHAPIRO
Keyword(s):  
Prime Ideal ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Valuation Domain ◽  
Ring Extension ◽  
Infinite Products ◽  
Normal Pair ◽  
Zero Divisors ◽  
Pair Property ◽  
Nontrivial Zero

Results of Davis on normal pairs (R, T) of domains are generalized to (commutative) rings with nontrivial zero-divisors, particularly complemented rings. For instance, if T is a ring extension of an almost quasilocal complemented ring R, then (R, T) is a normal pair if and only if there is a prime ideal P of R such that T = R[P], R/P is a valuation domain and PT = P. Examples include sufficient conditions for the "normal pair" property to be stable under formation of infinite products and ⋈ constructions.

scholarly journals Pi-balanced torsion-free modules over a discrete valuation domain

2006 ◽  
Vol 295 (1) ◽  
pp. 269-288 ◽  
Author(s):  
David M. Arnold ◽  
K.M. Rangswamy ◽  
Fred Richman
Keyword(s):  
Valuation Domain ◽  
Discrete Valuation Domain ◽  
Free Modules ◽  
Torsion Free
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