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 \({\mathbb {M}}_2({\mathbb {Z}}_2)\).
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Acknowledgements
This research was supported by a grant (Grant Number 117F070) from TUBITAK of Turkey and a Discovery Grant (Grant Number RGPIN-2016-04706) from NSERC of Canada. Part of the work was carried out when Yiqiang Zhou was visiting Gazi University. He gratefully acknowledges the hospitality from the host institute. The authors are grateful to Professor A. Leroy for formatting this article in the journal’s style.
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Koşan, M.T., Zhou, Y. A class of rings with the 2-sum property. AAECC 32, 399–408 (2021). https://doi.org/10.1007/s00200-021-00490-y
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DOI: https://doi.org/10.1007/s00200-021-00490-y