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Published in:
30-01-2021 | Original Paper
A class of rings with the 2-sum property
Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 3/2021
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Abstract
Recall that a ring satisfies the 2-sum property if each of its elements is a sum of two units. Here a ring R is said to satisfy the binary 2-sum property if, for any a, b in R, there exists a unit u of R such that both \(a-u\) and \(b-u\) are units. A well-known result, due to Goldsmith, Pabst and Scot, states that a semilocal ring satisfies the 2-sum property iff it has no image isomorphic to \(\mathbb {Z}_2\). It is proved here that a semilocal ring satisfies the binary 2-sum property iff it has no image isomorphic to \({\mathbb {Z}}_2\) or \({\mathbb {Z}}_3\) or \({\mathbb {M}}_2({\mathbb {Z}}_2)\).