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 / 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).