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Rings generated by their regular elements

Published online by Cambridge University Press:  18 May 2009

A. W. Chatters  and
S. M. Ginn
A. W. Chatters
Affiliation:
School of Mathematics, University Walk, Bristol Bs8 1Tw
S. M. Ginn
Affiliation:
Mathematics Department, Birkbeck College, Malet Street, London, Wc1
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The units of a ring R are defined by means of a multiplicative property, but in many cases they generate R additively. For example, it is shown in [5, Proposition 6] that if R is a semi-simple Artinian ring then every element of R is a sum of units if and only if the ring S = ℤ/2ℤ⊕ℤ/2ℤ is not a direct summand of R, where ℤ denotes the ring of integers. The theme of this paper is to investigate the corresponding situation concerning regular elements, i. e. elements which are not zero-divisors. We show that if R is a semi-prime right Goldie ring then every element of R is a sum of regular elements if and only if R does not have the ring S defined above as a direct summand (Corollary 2.9). We also characterise those Noetherian rings R such that every element of R is a sum of regular elements (Theorem 2. 6). The characterisation is in terms of the nature of certain prime factor rings of R, and it is again the presence of the ring S, this time in a particular way as a factor ring of R, which prevents R from being generated by its regular elements. If R has no non-zero Artinian one-sided ideals or if 2 is a regular element of R, then every element of R is a sum of regular elements (Corollaries 2. 5 and 2. 7). As an application we show in Section 3 that, for many Noetherian rings R, the set of elements of R which are divisible by every regular element of R is a two-sided ideal of R.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 25 , Issue 1 , January 1984 , pp. 1 - 5
Copyright
Copyright © Glasgow Mathematical Journal Trust 1984

References

REFERENCES

1.Chatters, A. W., A note on Noetherian orders in Artinian rings, Glasgow Math. J. 20 (1979), 125–128.CrossRefGoogle Scholar
2.Chatters, A. W. and Hajarnavis, C. R., Rings with chain conditions (Pitman, 1980).Google Scholar
3.Chatters, A. W., Divisible ideals of Noetherian rings, Bull. London Math. Soc. 13 (1981), 328–330.CrossRefGoogle Scholar
4.Henriksen, M., Two classes of rings generated by their units, J. Algebra 31 (1974), 182–193.CrossRefGoogle Scholar
5.Raphael, R., Rings which are generated by their units. J. Algebra 28 (1974), 199–205.CrossRefGoogle Scholar
6.Small, L. W. and Stafford, J. T., Regularity of zero divisors, Proc. London Math. Soc. (3) 44 (1982), 405–419.CrossRefGoogle Scholar
7.Stafford, J. T., Noetherian full quotient rings, Proc. London Math. Soc. (3) 44 (1982), 385–404.CrossRefGoogle Scholar