純粋・応用数学（含むガロア理論）3

[過去ﾛｸﾞ] 純粋・応用数学（含むガロア理論）3 (1002ﾚｽ)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

ﾘﾛｰﾄﾞ規制です｡10分ほどで解除するので､他のﾌﾞﾗｳｻﾞへ避難してください｡

626: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/23(日) 16:06:25.22 ID:ehdjUjVy

>>614
>行列環 Mn(R)で、零因子を含むヤコブソン根基（>>604）J(Mn(R)を作って
>商環 Mn(R)/J(Mn(R)) 作れば J(Mn(R)/J(Mn(R))) = {0} が言えて（>>605）
>零因子を含まない環が、できるのか

これも撤回（＾＾；
上記の話は、可換環 R の話みたい（>>619-620ご参照）

行列環が、Division ringになる条件
うん、これか
"Relation to fields and linear algebra
In fact the converse is also true and this gives a characterization of division rings via their module category: A unital ring R is a division ring if and only if every R-module is free.[7]"
（ unitary ring、単位的環、単位環 ）
むずいｗ（＾＾；

https://en.wikipedia.org/wiki/Division_ring
Division ring

Relation to fields and linear algebra
All fields are division rings; more interesting examples are the non-commutative division rings. The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring. In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring;[6] every division ring arises in this fashion from some simple module.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595166668/626

627: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/23(日) 16:07:04.45 ID:ehdjUjVy

>>626
つづき

Much of linear algebra may be formulated, and remains correct, for modules over a division ring D instead of vector spaces over a field. Doing so it must be specified whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. Working in coordinates, elements of a finite dimensional right module can be represented by column vectors, which can be multiplied on the right by scalars, and on the left by matrices (representing linear maps); for elements of a finite dimensional left module, row vectors must be used, which can be multiplied on the left by scalars, and on the right by matrices. The dual of a right module is a left module, and vice versa. The transpose of a matrix must be viewed as a matrix over the opposite division ring Dop in order for the rule (AB)^T = B^TA^T to remain valid.

Every module over a division ring is free; i.e., has a basis, and all bases of a module have the same number of elements. Linear maps between finite-dimensional modules over a division ring can be described by matrices; the fact that linear maps by definition commute with scalar multiplication is most conveniently represented in notation by writing them on the opposite side of vectors as scalars are. The Gaussian elimination algorithm remains applicable. The column rank of a matrix is the dimension of the right module generated by the columns, and the row rank is dimension of the left module generated by the rows; the same proof as for the vector space case can be used to show that these ranks are the same, and define the rank of a matrix.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595166668/627

642: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/24(月) 07:16:54.23 ID:+oiN9Lqm

>>626 補足
>https://en.wikipedia.org/wiki/Division_ring
>Division ring
>"Relation to fields and linear algebra
>In fact the converse is also true and this gives a characterization of division rings via their module category: A unital ring R is a division ring if and only if every R-module is free.[7]"
>（ unital/unitary ring、単位的環、単位環 ）
>斜体であるという性質は加群の圏の性質から特徴づけることもできる。環 R が斜体である必要十分条件はすべての左 R 加群が自由加群であることである[5]。

下記が参考になりそう
https://www.math.uni-bielefeld.de/~wcrawley/1617noncommalg/Noncommutative%20algebra.pdf
Noncommutative algebra
Bielefeld University, Winter Semester 2016/17
William Crawley-Boevey

https://www.ams.org/journals/tran/2002-354-05/S0002-9947-02-02927-6/S0002-9947-02-02927-6.pdf
CONSTRUCTING DIVISION RINGS
AS MODULE-THEORETIC DIRECT LIMITS
GEORGE M. BERGMAN
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 354, Number 5, Pages 2079?2114 published on January 8, 2002

Abstract. If R is an associative ring, one of several known equivalent types
of data determining the structure of an arbitrary division ring D generated by
a homomorphic image of R is a rule putting on all free R-modules of finite rank
matroid structures (closure operators satisfying the exchange axiom) subject
to certain functoriality conditions. This note gives a new description of how
D may be constructed from this data. (A classical precursor of this is the
construction of Q as a field with additive group a direct limit of copies of Z.)
The division rings of fractions of right and left Ore rings, the universal
division ring of a free ideal ring, and the concept of a specialization of division
rings are then interpreted in terms of this construction.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1595166668/642

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 0.035s