[過去ﾛｸﾞ] 純粋・応用数学（含むガロア理論）3 (1002ﾚｽ)
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626(4): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/23(日)16:06 ID:ehdjUjVy(7/16) AAS
>>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
省10
627(2): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/23(日)16:07 ID:ehdjUjVy(8/16) AAS
>>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
632: 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/23(日)16:13 ID:ehdjUjVy(13/16) AAS
>>626 追加訂正
（ unitary ring、単位的環、単位環 ）
↓
（ unital/unitary ring、単位的環、単位環 ）
でした（＾＾；
634: 2020/08/23(日)16:28 ID:7NMituVg(8/11) AAS
>>626-630
わけもわからず体にこだわる高卒素人🐎🦌ｗｗｗｗｗｗｗ
毛深い野獣の貴様には数学は無理だから諦めろ
642(2): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/24(月)07:16 ID:+oiN9Lqm(1/11) AAS
>>626 補足
>外部ﾘﾝｸ:en.wikipedia.org
>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]。
省23
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