Rings whose subrings are all Noetherian or Artinian

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 Prüfer $p$-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.