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Structure of Certain Periodic Rings

Published online by Cambridge University Press:  20 November 2018

Hazar Abu-Khuzam  and
Adil Yaqub
Hazar Abu-Khuzam
Affiliation:
Department of Mathematics, University of Petroleum and MineralsDhahran, Saudi Arabia
Adil Yaqub
Affiliation:
Department of Mathematics, University of CaliforniaSanta Barbara, CA 93106
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Abstract

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Let R be a periodic ring, N the set of nilpotents, and D the set of right zero divisors of R. Suppose that (i) N is commutative, and (ii) every x in R can be uniquely written in the form x = e + a, where e2 = e and a ∊ N. Then N is an ideal in R and R/N is a Boolean ring. If (i) is satisfied but (ii) is now assumed to hold merely for those elements x ∊ D, and if 1 ∊ R, then N is still an ideal in R and R/N is a subdirect sum of fields. It is further shown that if (i) is satisfied but (ii) is replaced by: "every right zero divisor is either nilpotent or idempotent," and if 1 ∊ R, then N is still an ideal in R and R/N is either a Boolean ring or a field.

Keywords

16A3816A70
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 1 , 01 March 1985 , pp. 120 - 123
Copyright
Copyright © Canadian Mathematical Society 1985

References

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