Abstract We present the conditions for a block matrix of a ring to have the image-kernel ( p , q ) {(p,q)} -inverse in the generalized Banachiewicz–Schur form. We give representations for the image-kernel inverses of the sum and the product of two block matrices. Some characterizations of the image-kernel ( p , q ) {(p,q)} -inverse in a ring with involution are investigated too.

中文翻译：

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环中块矩阵的图像核 (p,q) 逆的表示

摘要 我们提出了环的块矩阵在广义 Banachiewicz-Schur 形式中具有图像核 ( p , q ) {(p,q)} 逆的条件。我们给出了两个块矩阵的和和乘积的图像核逆的表示。还研究了具有对合的环中图像核 ( p , q ) {(p,q)} -逆的一些特征。