Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an integral domain. Now also assume that distinct ideals have distinct index; then is it true that $R$ is a UFD, or at least normal (integrally closed in its fraction field) ?
(note: $\dim R \le 1$ , now if $\dim R=0$, then $R$ is an Artinian domain, so a field. So w.l.o.g., assume $R$ has dimension $1$ )
