## Abstract

Let R be an integral domain and I a nonzero ideal of R. An ideal J ⊆ I is a t-reduction of I if (JI^{n})_{t} = (I^{n+1})_{t} for some positive integer n; and I is t-basic if it has no t-reduction other than the trivial ones. This paper investigates t-reductions of ideals in pullback constructions of type □. Section 2 examines the correlation between the notions of reduction and t-reduction in pseudo-valuation domains. Section 3 solves an open problem on whether the finite t-basic and υ-basic 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 t-basic ideal property to pullbacks in line with the result in Fontana-Gabelli, 1996, on PυMDs and the result in Gabelli-Houston, 1997, on υ-domains. This allows us to enrich the literature with new families of examples, which put the class of domains subject to the finite t-basic ideal property strictly between the two classes of υ-domains and integrally closed domains.

Original language | English |
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Pages (from-to) | 875-889 |

Number of pages | 15 |

Journal | New York Journal of Mathematics |

Volume | 22 |

State | Published - 2016 |

### Bibliographical note

Publisher Copyright:© 2016, University at Albany. All rights reserved.

## Keywords

- Basic ideal
- Integrally closed domain
- Pseudo-valuation domain
- Pullback
- PυMD
- Reduction of an ideal
- t-basic ideal
- t-operation
- t-reduction of an ideal
- υ-domain

## ASJC Scopus subject areas

- Mathematics (all)