[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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628(2): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/23(日)16:07 ID:ehdjUjVy(9/16) AAS
>>627
つづき

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]

The center of a division ring is commutative and therefore a field.[8] Every division ring is therefore a division algebra over its center. Division rings can be roughly classified according to whether or not they are finite-dimensional or infinite-dimensional over their centers. The former are called centrally finite and the latter centrally infinite. Every field is, of course, one-dimensional over its center. The ring of Hamiltonian quaternions forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.

外部ﾘﾝｸ:ja.wikipedia.org
斜体 (数学) division ring
斜体（しゃたい、英: skew field; 歪体, 独: Schiefkorper, 仏: corps, corps gauche）は加減乗除が可能な代数系である[1][注 1]。除法の可能な環であるという意味で可除環（かじょかん、division ring, Divisionsring）ともいう[3]。係数環を持ち、多元環の構造を持つことを強調する場合は、特に多元体[4]（たげんたい、division algebra, algebre a division; 可除多元環）と呼称することも多い[注 2]。非可換な積を持つ体を非可換体（ひかかんたい、non-commutative field, corps non commutatif）という[2]。
省3

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629(2): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/08/23(日)16:08 ID:ehdjUjVy(10/16) AAS
>>628

つづき

外部ﾘﾝｸ:en.wikipedia.org
Matrix ring
・The algebra M2(R) of 2 × 2 real matrices, which is isomorphic to the split-quaternions, is a simple example of a non-commutative associative algebra. Like the quaternions, it has dimension 4 over R, but unlike the quaternions, it has zero divisors, as can be seen from the following product of the matrix units: E11E21 = 0, hence it is not a division ring. Its invertible elements are nonsingular matrices and they form a group, the general linear group GL(2, R).

Structure
・In general, every semisimple ring is isomorphic to a finite direct product of full matrix rings over division rings, which may have differing division rings and differing sizes. This classification is given by the Artin?Wedderburn theorem.
省8

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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

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