This paper explains that the definition of Moore–Penrose inverses in a given algebra A does not at all require any matrix representation of A. The pseudo-inverses y of a given element x of A (such that \(xyx=x\) and \(yxy=y\)) involve the right ideals complementary to xA and the left ideals complementary to Ax; the Moore–Penrose inverse corresponds to the complementary right and left ideals selected by means of a positive involution on A. This paper is also the occasion to take the stock of several useful concepts: semi-simple rings, involutions of algebras, especially positive involutions.

中文翻译：

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没有矩阵的伪逆

本文解释说，给定代数A中Moore-Penrose逆的定义完全不需要A的任何矩阵表示。伪逆ý给定元素的X的甲（使得\（XYX = X \）和\（YXY = Y \） ）涉及的权利理想互补XA和左理想互补斧; Moore-Penrose逆与通过A的正对合选择的左右互补理想相对应。本文也是总结几个有用概念的机会：半简单环，代数的对合，尤其是正对合。