Abstract
In this paper, the notions of invariance and parallel sums as defined by Anderson and Duffin for matrices [Series and parallel addition of matrices, J. Math. Anal. Appl. 26 (1969) 576–594] are generalized to von Neumann regular rings.
Highlights
Let R be a von Neumann regular algebra over any commutative ring
B ∈ R, we set a− = {x ∈ R | axa = a}, and define the parallel sum P (a, b) of a, b as P (a, b) = a(a + b)−b. This notion, introduced by Anderson and Duffin using the Moore–Penrose inverses, arose from the notion of the impedance matrix of two n-port electrical networks connected in parallel [2]
The concept of parallel summability was extended by Rao and Mitra who proved similar results of Anderson and Duffin in a general setting replacing the Moore– Penrose inverse by a generalized inverse, known as inner inverse [7]
Summary
B ∈ R, we set a− = {x ∈ R | axa = a}, and define the parallel sum P (a, b) of a, b as P (a, b) = a(a + b)−b. The concept of parallel summability was extended by Rao and Mitra who proved similar results of Anderson and Duffin in a general setting replacing the Moore– Penrose inverse by a generalized inverse, known as inner inverse [7]. This is an Open Access article published by World Scientific Publishing Company. In an earlier paper [1], we proved that in any von Neumann regular ring R, if such an element c exists it must be unique.
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