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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The additive group of commutative rings generated by idempotents


Author: Paul Hill
Journal: Proc. Amer. Math. Soc. 38 (1973), 499-502
MSC: Primary 13A99
DOI: https://doi.org/10.1090/S0002-9939-1973-0316439-2
MathSciNet review: 0316439
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Abstract: If $ R$ is a ring, let $ {R^ + }$ denote its additive group. Our purpose is to give an elementary proof that if $ R$ is a commutative ring generated by idempotents, then any subring of $ R$ generated by idempotents is pure. This yields immediately an independent proof of the following result of G. M. Bergman. If $ R$ is a commutative ring with identity and if $ R$ is generated by idempotents, then $ {R^ + }$ is a direct sum of cyclic groups.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0316439-2
Keywords: Specker theorem, Nöbeling theorem, direct sum of cyclic groups, pure subgroups, commutative rings, idempotent generators
Article copyright: © Copyright 1973 American Mathematical Society