Assume $R$ is a commutative Noetherian ring of finite Krull dimension; $R'$ is a not commutative ring that contains $R$ in its center and also finitely generated as an $R$-module.
If the (left) global dimension of $R'$ is infinite, should there exist a finitely generated $R'$-module whose projective dimension is infinite? Is this clear if $R$ is regular?
