Transfer of the GPIT Property in Pullbacks

Transfer of the GPIT Property in Pullbacks

Let 𝑇 be a commutative ring, 𝐼 a prime ideal of 𝑇, 𝐷 a subring of 𝑇/𝐼, and 𝑅 the pullback 𝑇×𝑇/𝐼𝐷. Ascent and descent results are given for the transfer of the 𝑛-PIT and GPIT (generalized principal ideal theorem) properties between 𝑇 and 𝑅. As a consequence, it follows that if 𝐼 is a maximal ideal of...

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Main Authors: David E. Dobbs, Jay Shapiro
Format: Article
Language:English
Published: Hindawi Limited 2012-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Online Access:http://dx.doi.org/10.1155/2012/743873
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spelling doaj-38bb38c1f1954d5e8758213855d818ae2020-11-25T00:13:09ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252012-01-01201210.1155/2012/743873743873Transfer of the GPIT Property in PullbacksDavid E. Dobbs0Jay Shapiro1Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1320, USADepartment of Mathematics, George Mason University, Fairfax, VA 22030-4444, USALet 𝑇 be a commutative ring, 𝐼 a prime ideal of 𝑇, 𝐷 a subring of 𝑇/𝐼, and 𝑅 the pullback 𝑇×𝑇/𝐼𝐷. Ascent and descent results are given for the transfer of the 𝑛-PIT and GPIT (generalized principal ideal theorem) properties between 𝑇 and 𝑅. As a consequence, it follows that if 𝐼 is a maximal ideal of both 𝑇 and 𝑅, then 𝑅 satisfies 𝑛-PIT (resp., GPIT) if and only if 𝑇 satisfies 𝑛-PIT (resp., GPIT).http://dx.doi.org/10.1155/2012/743873
collection DOAJ
language English
format Article
sources DOAJ
author David E. Dobbs
Jay Shapiro
spellingShingle David E. Dobbs
Jay Shapiro
Transfer of the GPIT Property in Pullbacks
International Journal of Mathematics and Mathematical Sciences
author_facet David E. Dobbs
Jay Shapiro
author_sort David E. Dobbs
title Transfer of the GPIT Property in Pullbacks
title_short Transfer of the GPIT Property in Pullbacks
title_full Transfer of the GPIT Property in Pullbacks
title_fullStr Transfer of the GPIT Property in Pullbacks
title_full_unstemmed Transfer of the GPIT Property in Pullbacks
title_sort transfer of the gpit property in pullbacks
publisher Hindawi Limited
series International Journal of Mathematics and Mathematical Sciences
issn 0161-1712
1687-0425
publishDate 2012-01-01
description Let 𝑇 be a commutative ring, 𝐼 a prime ideal of 𝑇, 𝐷 a subring of 𝑇/𝐼, and 𝑅 the pullback 𝑇×𝑇/𝐼𝐷. Ascent and descent results are given for the transfer of the 𝑛-PIT and GPIT (generalized principal ideal theorem) properties between 𝑇 and 𝑅. As a consequence, it follows that if 𝐼 is a maximal ideal of both 𝑇 and 𝑅, then 𝑅 satisfies 𝑛-PIT (resp., GPIT) if and only if 𝑇 satisfies 𝑛-PIT (resp., GPIT).
url http://dx.doi.org/10.1155/2012/743873
work_keys_str_mv AT davidedobbs transferofthegpitpropertyinpullbacks
AT jayshapiro transferofthegpitpropertyinpullbacks
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