まあね。しかし、「行列式det(Ａ)が逆元を持つこと」ことが本質なんだよ（下記の通りだ） (参考) >>153より再録 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.
