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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 Depth of noncommutative subrings ： ウィキペディア英語版 Depth of noncommutative subrings In ring theory and Frobenius algebra extensions, fields of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. The notion of depth two is important in a certain noncommutative Galois theory, which generates Hopf algebroids in place of the more classical Galois groups, whereas the notion of depth greater than two measures the defect, or distance, from being depth two in a tower of iterated endomorphism rings above the subring. A more recent definition of depth of any unital subring in any associative ring is proposed (see below) in a paper studying the depth of a subgroup of a finite group as group algebras over a commutative ring. == Definition and first examples == A unital subring $B\; \backslash subseteq\; A$ has (or is) right depth two if there is a split epimorphism of natural A-B-bimodules from $A^n\; \backslash rightarrow\; A\; \backslash otimes\_B\; A$ for some positive integer n; by switching to natural B-A-bimodules, there is a corresponding definition of left depth two. Here we use the usual notation $A^n\; =\; A\; \backslash times\; \backslash ldots\; \backslash times\; A$ (n times) as well as the common notion, p is a split epimorphism if there is a homomorphism q in the reverse direction such that pq = identity on the image of p. (Sometimes the subring B in A is referred to as the ring extension A over B; the theory works as well for a ring homomorphism B into A, which induces right and left B-modules structures on A.) Equivalently, the condition for left or right depth two may be given in terms of a split monomorphism of bimodules where the domains and codomains above are reversed. For example, let A be the group algebra of a finite group G (over any commutative base ring k; see the articles on group theory and group ring for the elementary definitions). Let B be the group (sub)algebra of a normal subgroup H of index n in G with coset representatives $g\_1,\backslash cdots,g\_n$. Define a split A-B epimorphism p:$A^n\; \backslash rightarrow\; A\; \backslash otimes\_B\; A$ by $p((a\_1,\backslash cdots,a\_n))\; =\; \backslash sum\_^n\; a\_i\; g\_i^\; \backslash otimes\_B\; g\_i$. It is split by the mapping $q:\; A\; \backslash otimes\_B\; A\; \backslash rightarrow\; A^n$ defined by $q(a\; \backslash otimes\_B\; a\text{'})\; =\; (a\; \backslash gamma\_1(a\text{'}),\backslash cdots,a\backslash gamma\_n(a\text{'}))$ where $\backslash gamma\_i(g)\; =\; \backslash delta\_\; g$ for g in the coset $g\_jH$ (and extended linearly to a mapping A into B, a B-B-module homomorphism since H is normal in G): the splitting condition pq = the identity on $A\; \backslash otimes\_B\; A$ is satisfied. Thus B is right depth two in A. As another example (perhaps more elementary than the first; see ring theory or module theory for some of the elementary notions), let A be an algebra over a commutative ring B, where B is taken to be in the center of A. Assume A is a finite projective B-module, so there are B-linear mapping $f\_i:\; A\; \backslash rightarrow\; B$ and elements $x\_i\; \backslash in\; A$ (i = 1,...,n) called a projective base for the B-module A if it satisfies $\backslash sum\_^n\; x\_i\; f\_i(a)\; =\; a$ for all a in A. It follows that B is left depth two in A by defining $p(a\_1,\backslash cdots,a\_n)\; =\; \backslash sum\_^n\; x\_i\; \backslash otimes\_B\; a\_i$ with splitting map $q(a\; \backslash otimes\_B\; a\text{'})\; =\; (f\_1(a)a\text{'},\backslash cdots,f\_n(a)a\text{'})$ as the reader may verify. A similar argument naturally shows that B is right depth two in A. 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「Depth of noncommutative subrings」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.