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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Generic matrix ring ： ウィキペディア英語版 Generic matrix ring In algebra, a generic matrix ring of size ''n'' with variables $X\_1,\; \backslash dots\; X\_m$, denoted by $F\_n$, is a sort of a universal matrix ring. It is universal in the sense that, given a commutative ring ''R'' and ''n''-by-''n'' matrices $A\_1,\; \backslash dots,\; A\_m$ over ''R'', any mapping $X\_i\; \backslash mapsto\; A\_i$ extends to the ring homomorphism (called evaluation) $F\_n\; \backslash to\; M\_n(R)$. Explicitly, given a field ''k'', it is the subalgebra $F\_n$ of the matrix ring $M\_n(k(\backslash le\; l\; \backslash le\; m,\; 1\; \backslash le\; i,\; j\; \backslash le\; n))$ generated by ''n''-by-''n'' matrices $X\_1,\; \backslash dots,\; X\_m$, where $(X\_l)\_$ are matrix entries and commute by definition. For example, if ''m'' = 1, then $F\_1$ is a polynomial ring in one variable. For example, a central polynomial is an element of the ring $F\_n$ that will map to a central element under an evaluation. (In fact, it is in the invariant ring $k()^$ since it is central and invariant.) By definition, $F\_n$ is a quotient of the free ring $k\backslash langle\; t\_1,\; \backslash dots,\; t\_m\; \backslash rangle$ with $t\_i\; \backslash mapsto\; X\_i$ by the ideal consisting of all ''p'' that vanish identically on any ''n''-by-''n'' matrices over ''k''. The universal property means that any ring homomorphism from $k\backslash langle\; t\_1,\; \backslash dots,\; t\_m\; \backslash rangle$ to a matrix ring factors through $F\_n$. This has a following geometric meaning. In algebraic geometry, the polynomial ring $k(\backslash dots,\; t\_m)$ is the coordinate ring of the affine space $k^m$ and to give a point of $k^m$ is to give a ring homomorphism (evaluation) $k(\backslash dots,\; t\_m)\; \backslash to\; k$ (either by the Hilbert nullstellensatz or by the scheme theory). The free ring $k\backslash langle\; t\_1,\; \backslash dots,\; t\_m\; \backslash rangle$ plays the role of the coordinate ring of the affine space in the noncommutative algebraic geometry (i.e., we don't demand free variables to commute) and thus a generic matrix ring of size ''n'' is the coordinate ring of a noncommutative affine variety whose points are the Spec's of matrix rings of size ''n'' (see below for a more concrete discussion.) == The maximum spectrum of a generic matrix ring == For simplicity, assume ''k'' is algebraically closed. Let ''A'' be an algebra over ''k'' and let $\backslash operatorname\_n(A)$ denote the set of all maximal ideals $\backslash mathfrak$ in ''A'' such that $A/\backslash mathfrak\; \backslash approx\; M\_n(k)$. If ''A'' is commutative, then $\backslash operatorname\_1(A)$ is the maximum spectrum of ''A'' and $\backslash operatorname\_n(A)$ is empty for any $n\; >\; 1$. 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「Generic matrix ring」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.