Local Ring Equivalent Conditions

If R is a commutative ring with 1 then the following conditions are equivalent.
(i) R is a local ring, that is, a commutative ring with 1 which has a unique maximal ideal.
(ii) All nonunits of R are contained in some ideal M\neq R.
(iii) The nonunits of R form an ideal.

(H pg 147)

(i)\implies(ii): If a\in R is a nonunit, then (a)\neq R since 1\notin (a). Therefore (a) (and hence a) is contained in the unique maximal ideal M of R, since M must contain every ideal of R (except R itself).

(ii)\implies(iii): Let S be the set of all nonunits of R. We have S\subseteq M\neq R. Let x\in M. Since M\neq R, x cannot be a unit. So x\in S. Thus M\subseteq S. Hence S=M, which is an ideal.

(iii)\implies(i): Assume S, the set of nonunits, form an ideal. Let I\neq R be a maximal ideal. Let a\in I, then a cannot be a unit so a\in S. Thus I\subseteq S\neq R. By maximality S=I and this shows S is the unique maximal ideal.

Author: mathtuition88


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