A change of rings theorem and the Artin-Rees property

Abstract:

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