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In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring .〔Isaacs, p. 184〕 The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vector space.〔Such rings of linear transformations are also known as full linear rings.〕〔Isaacs, Corollary 13.16, p. 187〕 This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson.〔(Jacobson, Nathan "Structure Theory of Simple Rings Without Finiteness Assumptions" )〕 This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple Artinian rings. ==Motivation and formal statement== Let be a ring and let be a simple right -module. If is a non-zero element of , (where is the cyclic submodule of generated by ). Therefore, if are non-zero elements of , there is an element of that induces an endomorphism of transforming to . The natural question now is whether this can be generalized to arbitrary (finite) tuples of elements. More precisely, find necessary and sufficient conditions on the tuple and separately, so that there is an element of with the property that for all . If is the set of all -module endomorphisms of , then Schur's lemma asserts that is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the are linearly independent over . With the above in mind, the theorem may be stated this way: :The Jacobson Density Theorem. Let be a simple right -module, , and a finite and -linearly independent set. If is a -linear transformation on then there exists such that for all in .〔Isaacs, Theorem 13.14, p. 185〕 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「Jacobson density theorem」の詳細全文を読む スポンサード リンク
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