If is a commutative ring with 1 then the following conditions are equivalent.
(i) is a local ring, that is, a commutative ring with 1 which has a unique maximal ideal.
(ii) All nonunits of are contained in some ideal .
(iii) The nonunits of form an ideal.
(H pg 147)
(i)(ii): If is a nonunit, then since . Therefore (and hence ) is contained in the unique maximal ideal of , since must contain every ideal of (except itself).
(ii)(iii): Let be the set of all nonunits of . We have . Let . Since , cannot be a unit. So . Thus . Hence , which is an ideal.
(iii)(i): Assume , the set of nonunits, form an ideal. Let be a maximal ideal. Let , then cannot be a unit so . Thus . By maximality and this shows is the unique maximal ideal.