|
In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring ''R'' for which every simple right ''R'' module is isomorphic to a right ideal of ''R''.〔This ideal is necessarily a minimal right ideal.〕 Analogously the notion of a left Kasch ring is defined, and the two notions are independent of each other. Kasch rings are named in honor of mathematician Friedrich Kasch. Kasch originally called Artinian rings whose proper ideals have nonzero annihilators ''S-rings''. The characterizations below show that Kasch rings generalize S-rings. ==Definition== Equivalent definitions will be introduced only for the right-hand version, with the understanding that the left-hand analogues are also true. The Kasch conditions have a few equivalences using the concept of annihilators, and this article uses the same notation appearing in the annihilator article. In addition to the definition given in the introduction, the following properties are equivalent definitions for a ring ''R'' to be right Kasch. They appear in : # For every simple right ''R'' module ''S'', there is a nonzero module homomorphism from ''M'' into ''R''. # The maximal right ideals of ''R'' are right annihilators of ring elements, that is, each one is of the form where ''x'' is in ''R''. # For any maximal right ideal ''T'' of ''R'', . # For any proper right ideal ''T'' of ''R'', . # For any maximal right ideal ''T'' of ''R'', . # ''R'' has no dense right ideals except ''R'' itself. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kasch ring」の詳細全文を読む スポンサード リンク
|