Some Remarks on One-Sided Inverses
1989; Linguagem: Inglês
10.1007/978-1-4612-3694-8_6
Autores Tópico(s)Matrix Theory and Algorithms
ResumoLet $$ \mathfrak{A} $$ be an arbitrary ring with an identity 1, and suppose that $$ \mathfrak{A} $$ contains a pair of elements u, v such that 1 $$ uv = 1\;but\;vu \ne 1 $$ We introduce the elements 2 $$ {e_{i,j}} = {v^{i - 1}}{u^{j - 1}} - {v^i}{u^j} $$ 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: 3 $$ {e_{i,j}}{e_{rs}} = {\delta_{jr}}{e_{is}} $$ 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).
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