 

S. Kabbaj, A. Kadri, and A. Mimouni
On treductions of ideals in pullbacks view print


Published: 
August 18, 2016

Keywords: 
Reduction of an ideal, basic ideal, toperation, treduction of an ideal, tbasic ideal, pseudovaluation domain, pullback, PvMD, vdomain, integrally closed domain 
Subject: 
13A15, 13A18, 13F05, 13G05, 13C20 


Abstract
Let R be an integral domain and I a nonzero ideal of R. An ideal J⊆ I is a treduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some
positive integer n; and I is tbasic if it has no treduction other than the trivial ones. This paper investigates treductions of ideals in pullback constructions of type □. Section 2 examines the correlation between the notions of reduction and treduction in pseudovaluation domains. Section 3 solves an open problem on whether the finite tbasic and vbasic ideal properties are distinct. We prove that these two notions coincide in any arbitrary domain. Section 4 features the main result, which establishes the transfer of the finite tbasic ideal property to pullbacks in line with the result in FontanaGabelli, 1996, on PvMDs and the result in
GabelliHouston, 1997, on vdomains. This allows us to enrich the literature with new families of examples, which put the class of domains subject to the finite tbasic ideal property strictly between the two classes of vdomains and integrally closed domains.


Acknowledgements
Supported by King Fahd University of Petroleum & Minerals under Research Grant # RG1328.


Author information
S. Kabbaj:
Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, KSA
kabbaj@kfupm.edu.sa
A. Kadri:
Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, KSA
g201004080@kfupm.edu.sa
A. Mimouni:
Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, KSA
amimouni@kfupm.edu.sa

