Wikipedia lists a few equivalent definitions of local rings, the first two of which are
- $R$ has a unique maximal left ideal.
- $R$ has a unique maximal right ideal.
However, it does not list this condition:
- $R$ has a unique maximal two-sided ideal.
I feel that both conditions in Wikipedia imply $R\backslash R^\times$ is an additive subgroup, thus implies 3. Then why is 3 not listed? Is it because 3 is not strong enough in general? What is a counterexample? Or is it simply "too easy to be listed?"
Best Answer
Yes, condition 3 is much weaker. For instance, if $R=M_n(k)$ for a field $k$, then $0$ is the unique maximal two-sided ideal, but one-sided maximal ideals in $R$ are not unique if $n>1$.