criterion for maximal ideal
Proof. . Let first be a maximal ideal of and
. Because , there exist some elements
and such that . Consequently, .
. Assume secondly that the ideal satisfies the condition (1). Now there must be a maximal ideal of such that
Let us make the antithesis that is non-empty. Choose an element
By our assumption, we can choose another element of such that
Then we have
which is impossible since with 1 the ideal would contain the whole . Thus the antithesis is wrong and is maximal.
|Title||criterion for maximal ideal|
|Date of creation||2013-03-22 19:10:40|
|Last modified on||2013-03-22 19:10:40|
|Last modified by||pahio (2872)|