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Unit sum numbers of right self-injective rings

Published online by Cambridge University Press:  17 April 2009

Dinesh Khurana  and
Ashish K. Srivastava
Dinesh Khurana
Affiliation:
Department of Mathematics, Panjab University, Chandigarh-160014, India e-mail: dkhurana@pu.ac.in
Ashish K. Srivastava
Affiliation:
Department of Mathematics, Ohio University, Athens, OH 45701, United States of America e-mail: ashish@math.ohiou.edu
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In a recent paper (which is to appear in J. Algebra Appl.) we proved that every element of a right self-injective ring R is a sum of two units if and only if R has no factor ring isomorphic to ℤ2 and hence the unit sum number of a nonzero right self-injective ring is 2, ω or ∞. In this paper we characterise right self-injective rings with unit sum numbers ω and ∞. We prove that the unit sum number of a right self-injective ring R is ω if and only if R has a factor ring isomorphic to ℤ2 but no factor ring isomorphic to ℤ2 × ℤ2, and also in this case every element of R is a sum of either two or three units. It follows that the unit sum number of a right self-injective ring R is ∞ precisely when R has a factor ring isomorphic to ℤ2 × ℤ2. We also answer a question of Henriksen (which appeared in J. Algebra, Question E, page 192), by giving a large class of regular right self-injective rings having the unit sum number ω in which not all non-invertible elements are the sum of two units.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 75 , Issue 3 , June 2007 , pp. 355 - 360
Copyright
Copyright © Australian Mathematical Society 2007

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