International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 743873, 6 pages
http://dx.doi.org/10.1155/2012/743873
Research Article

Transfer of the GPIT Property in Pullbacks

David E. Dobbs1 and Jay Shapiro2

1Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1320, USA
2Department of Mathematics, George Mason University, Fairfax, VA 22030-4444, USA

Received 6 September 2011; Accepted 15 November 2011

Academic Editor: Ayman Badawi

Copyright © 2012 David E. Dobbs and Jay Shapiro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑇 be a commutative ring, 𝐼 a prime ideal of 𝑇 , 𝐷 a subring of 𝑇 / 𝐼 , and 𝑅 the pullback 𝑇 × 𝑇 / 𝐼 𝐷 . 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).