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- W2005888868 abstract "Let ( mathfrak{A} ) be an arbitrary ring with an identity 1, and suppose that ( mathfrak{A} ) contains a pair of elements u, v such that $$ uv = 1;but;vu ne 1 $$ (1) We introduce the elements $$ {e_{i,j}} = {v^{i - 1}}{u^{j - 1}} - {v^i}{u^j} $$ (2) for i, j = 1, 2, 3, …, where it is understood that u 0 = l =v 0. It can be verified directly that the e ij thus defined satisfy the multiplication table for matrix units: $$ {e_{i,j}}{e_{rs}} = {delta_{jr}}{e_{is}} $$ (3) In particular the elements e i = e ij are orthogonal idempotent elements. No e ij =0. For by (3) the vanishing of one of the e ij implies the vanishing of all; in particular, it implies that $$ 0 = e_1 = 1 - vu $$ contrary to (1)." @default.
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- W2005888868 date "1989-01-01" @default.
- W2005888868 modified "2023-09-27" @default.
- W2005888868 title "Some Remarks on One-Sided Inverses" @default.
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- W2005888868 doi "https://doi.org/10.1007/978-1-4612-3694-8_6" @default.
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