Artinian rings and PID Let $R$ be a commutative ring with unity. Suppose that $R$ is a principal ideal domain, and $0\ne t\in R$. I'm trying to show that $R/Rt$ is an artinian $R$-module, and so is an artinian ring if $t$ is not a unit in $R$.I'm not sure how to begin. please help.

Hint: Show that the ideals of $R/Rt$ are all principal, and in fact, are in bijection with the divisors of $t$ in $R$ (considered up to multiplication by a unit). Then use  that $R$ is a UFD (since it's a PID). Finally, note that the ideals of $R/Rt$ are also precisely its $R$-submodules.
