Suppose $R$ is a commutative ring with identity of infinite cardinality. If for every non-zero ideal $I$ of $R$ we have $|\frac{R}{I}|<\infty$ then $R$ is an integral domain. How do I go on proving this?
Asked
Active
Viewed 82 times
3
-
1By "proper" you also mean to include "nonzero" apparently? "Proper" typically means "not the whole ring." – rschwieb Aug 22 '17 at 12:44
-
This might be of some help: https://projecteuclid.org/euclid.rmjm/1446472429 – Dirk Aug 22 '17 at 12:58
-
rschwieb, yes non-zero. as otherwise the condition can never be satisfied for |R/(0)| = |R| = infinity – Marcel S Aug 22 '17 at 15:59
-
2See https://math.stackexchange.com/q/126206. – Minseon Shin Aug 22 '17 at 17:54
-
Minseon Shin Thank you! I couldn't find this post when I was writing it unfortunately... – Marcel S Aug 24 '17 at 09:30