We give a constructive sufficient condition for a matrix over a commutative ring to be von Neumann regular, and we show that it is also necessary over local rings. Specifically, we prove that a matrix A over a local commutative ring is von Neumann regular if and only if A has an invertible $\rho (A)\times \rho (A)$-submatrix if and only if the determinantal rank $\rho (A)$ and the McCoy rank of A coincide. We deduce consequences to (products of local) commutative rings, and we determine the number of von Neumann regular matrices over some finite rings of residue classes and group algebras.

中文翻译：

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重温冯·诺依曼正则矩阵

我们给出了交换环上的矩阵为冯·诺依曼正则矩阵的建设性充分条件，并且证明了它在局部环上也是必要的。具体来说，我们证明局部交换环上的矩阵A是冯诺依曼正则当且仅当A具有可逆$\rho (A)\times \rho (A)$-子矩阵当且仅当行列式秩$\rho (A)$和A的 McCoy 等级一致。我们推导出（局部的）交换环的结果，并确定剩余类和群代数的一些有限环上的冯诺依曼正则矩阵的数量。