|
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero. == Definition == A ring ''R'' is said to be a left primitive ring if and only if it has a faithful simple left ''R''-module. A right primitive ring is defined similarly with right ''R''-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in . Another example found by Jategaonkar showing the distinction can be found in An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal containing no nonzero two-sided ideals. The analogous definition for right primitive rings is also valid. The structure of left primitive rings is completely determined by the Jacobson density theorem: A ring is left primitive if and only if it is isomorphic to a dense subring of the ring of endomorphisms of a left vector space over a division ring. Another equivalent definition states that a ring is left primitive if and only if it is a prime ring with a faithful left module of finite length (, (Ex. 11.19, p. 191 )). 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「primitive ring」の詳細全文を読む スポンサード リンク
|