Abstract
AbstractLet R be a semiprime ring with extended centroid C and let $I(x)$ denote the set of all inner inverses of a regular element x in R. Given two regular elements $a, b$ in R, we characterise the existence of some $c\in R$ such that $I(a)+I(b)=I(c)$ . Precisely, if $a, b, a+b$ are regular elements of R and a and b are parallel summable with the parallel sum ${\cal P}(a, b)$ , then $I(a)+I(b)=I({\cal P}(a, b))$ . Conversely, if $I(a)+I(b)=I(c)$ for some $c\in R$ , then $\mathrm {E}[c]a(a+b)^{-}b$ is invariant for all $(a+b)^{-}\in I(a+b)$ , where $\mathrm {E}[c]$ is the smallest idempotent in C satisfying $c=\mathrm {E}[c]c$ . This extends earlier work of Mitra and Odell for matrix rings over a field and Hartwig for prime regular rings with unity and some recent results proved by Alahmadi et al. [‘Invariance and parallel sums’, Bull. Math. Sci.10(1) (2020), 2050001, 8 pages] concerning the parallel summability of unital prime rings and abelian regular rings.
Highlights
Throughout, rings are associative, not necessarily with unity
If I(a) + I(b) = I(c) for some c ∈ R, E[c]a(a + b)−b is invariant for all (a + b)− ∈ I(a + b), where E[c] is the smallest idempotent in C satisfying c = E[c]c
We denote by Reg(R) the set of all regular elements in the ring R and by I(a) the set of all inner inverses of a in R
Summary
Throughout, rings are associative, not necessarily with unity. Given elements a, b in a ring R, the elements (1 − a)b and b(1 − a) always mean b − ab and b − ba, respectively. Abstract Let R be a semiprime ring with extended centroid C and let I(x) denote the set of all inner inverses of a regular element x in R. Keywords and phrases: abelian ring, extended centroid, inner inverse, parallel summable, regular element, semiprime (prime) ring, triplet invariance. When R is a semiprime ring with I(a) + I(b) = I(c), it follows from Theorem 2.8 below that c is uniquely determined.
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