Regularity and unit-regularity of generalized semigroups of linear transformations
Let \(V\) and \(W\) be vector spaces over a division ring \(D\), denote by \(L_D(V,W)\) the collection of all linear transformations from \(V\) to \(W\), and let \(\theta\) be a linear transformation from \(W\) to \(V\). Define an associative multiplication \(*\) on \(L_D(V,W)\) by \(\alpha*\beta=\alpha\theta\beta\) and denote this semigroup by \((L_D(V,W),\theta)\). The author proves that \((L_D(V,W),\theta)\) is regular if and only if \(V=\{0\}\), \(W=\{0\}\), or \(\theta\) is an isomorphism from \(W\) onto \(V\) and he proves that \((L_D(V,W),\theta)\) has an identity and is regular if and only if \(V=\{0\}\), \(W=\{0\}\), or \(\theta\) is an isomorphism from \(W\) onto \(V\) and \(\dim_DV<\infty\).
scientific article; zbMATH DE number 1836970