Transfer of the GPIT Property in Pullbacks

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/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).