Inter-universal geometry と ABC予想 (応援スレ) 71

[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 71 (1002ﾚｽ)
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153: １３２人目の素数さん [] 2024/04/25(木) 17:34:35.56 ID:G15DoGGk

>>152
>en.wikipediaではInvertible_matrix
>「リング上にはランクの概念が存在しない」

なるほど なるほど
de.wikipedia Reguläre Matrix では
ランクの概念が ”Equivalent characterizations （Äquivalente Charakterisierungen）”において
出てこないね

ふむふむ　；ｐ）
独語は、ドイツ留学の御大の出番かも

なお、独Reguläre Matrix、英Invertible matrix が、各国の数学用語で
正則行列はドイツ流ですな

(参考) （独原文は略す）
https://de.wikipedia.org/wiki/Regul%C3%A4re_Matrix
Reguläre Matrix
(google 独→英訳)
Equivalent characterizations （Äquivalente Charakterisierungen）
Regular matrices over a unitären kommutativen Ring(単位元を持つ可換環？)
More general is one (n×n)-Matrix  A with entries from a commutative ring with one
R invertible if and only if one of the following equivalent conditions is met:

・There is a matrix B with  AB=I=BA.
・The determinant of A is a unit in R (one also speaks of a unimodular matrix ).
・For all b ∈ R^{n} there is exactly one solution  x∈ R^{n} of the linear system of equations Ax=b.
・For all b ∈ R^{n} there is  at least one solution  x∈ R^{n} of the linear system of equations Ax=b.
・The row vectors form a basis of  R^{n}.
・Generate the row vectors R^{n}.
・The column vectors form a basis of R^{n}.
・Create the column vectors R^{n}.
・By A linear mapping described R^{n} → R^{n},x→ Ax, is surjective (or even bijective ).
・The transposed matrix A^{T} is invertible.

With a singular (n×n)-Matrix A with entries from a commutative ring with one R none of the above conditions are met.
The essential difference here compared to the case of a body is that, in general, the injectivity of a linear mapping no longer results in its surjectivity (and thus its bijectivity), as in the simple example
Z →Z, x→ 2x shows.

http://rio2016.5ch.net/test/read.cgi/math/1713536729/153

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