A change of rings theorem and the Artin-Rees property

Abstract

A two-sided ideal A \mathfrak {A} of the ring R R is said to have the left A R AR property if for every left ideal I I and every k k there exists an n n such that A n ∩ I ⊂ A k I {\mathfrak {A}^n} \cap I \subset {\mathfrak {A}^k}I . Let R R be a left noetherian ring and A \mathfrak {A} a two-sided ideal contained in its Jacobson radical. If A \mathfrak {A} has the AR \operatorname {AR} property then l gl dim R ⩽ p dim R / A + l gl dim  R / A {\text {l}}\;{\text {gl}}\;{\text {dim}}\;R \leqslant {\text {p}}\;{\text {dim}}\;R/\mathfrak {A} + {\text {l}}\;{\text {gl dim }}R/\mathfrak {A} , where p dim R / A {\text {p}}\;{\text {dim}}\;R/\mathfrak {A} denotes the (left) projective dimension of the module R / A R/\mathfrak {A} .