Decompositions of matrices and linear transformations

Abstract

The aim of this thesis is to discuss how to express a matrix (or a linear transformation) as the sum of two invertible matrices (or invertible linear transformations) with some constraints. The work for this thesis is two-fold. Firstly, it is proved that if R is a semilocal ring or an exchange ring with primitive factors Artinian then R satisfies the Goodearl-Menal condition if and only if no homomorphic images of R is isomorphic to either Z₂ or Z₃ or M₂ (Z₂). These results correct two existing results in the literature. Secondly, for the ring R of linear transformations of a right vector space over a division ring D, two results are proved in this thesis: (1) If |D| > 3, then for any a ∈ R there exists a unit u of R such that both a + u and a - u⁻¹ are units of R; (2) If |D| > 2, then for any a ∈ R there exists a unit u of R such that both a - u and a - u⁻¹ are units of R. Result (1) extends a result of H. Chen [7] that the ring of linear transformations of a countably generated right vector space over a division ring D with |D| > 3 satisfies the condition that for any a ∈ R, there exists u ∈ U (R) such that a + u and a - u⁻¹ ∈ U (R). And result (2) answers a question raised by H. Chen [7] whether the ring of linear transformations of a countable generated right vector space over a division ring D with |D| > 2 satisfies the condition that for any a ∈ R, there exists u ∈ U (R) such that a - u and a - u⁻¹ ∈ U (R). Connections of these conditions with some well-known conditions in ring theory are also discussed.