  We study noncommutative rings whose proper subrings all satisfy the same
chain condition. We show that if every proper subring of a ring $R$ is right
Noetherian, then $R$ is either right Noetherian or the trivial extension of
$\mathbb{Z}$ by the Pr\"ufer $p$-group for a prime $p$. We also prove that if
every proper subring of $R$ is right Artinian, then $R$ is either right
Artinian or $\mathbb{Z}$. For commutative rings, both results were proved by
Gilmer and Heinzer in 1992. Our result for right Artinian subrings only
generalises the absolute case of their commutative result. We generalise the
full result (when only certain subrings are right Artinian) in the context of
PI rings.
