valuation domain is local
Let and first be such elements of that is a unit of ; we may suppose that since otherwise one of and is instantly stated to be a unit. Because is a valuation domain in , therefore e.g. . Because now and belong to , so does also the product , i.e. is a unit of . We can conclude that the difference must be a non-unit whenever and are non-units.
Let and then be such elements of that is its unit, i.e. . Now we see that
and consequently and both are units. So we conclude that the product must be a non-unit whenever is an element of and is a non-unit.
Thus the non-units form an ideal . Suppose now that there is another ideal of such that . Since contains all non-units, we can take a unit in . Thus also the product , i.e. 1, belongs to , or . So we see that is a maximal ideal. On the other hand, any maximal ideal of contains no units and hence is contained in ; therefore is the only maximal ideal.
|Title||valuation domain is local|
|Date of creation||2013-03-22 14:54:49|
|Last modified on||2013-03-22 14:54:49|
|Last modified by||pahio (2872)|