unit sum number
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2011 ◽  
Vol 07 (03) ◽  
pp. 635-644
Author(s):  
VOLKER ZIEGLER

We consider a variation of the unit sum number problem for quadratic fields and prove various results.


2009 ◽  
Vol 93 (3) ◽  
pp. 259-268 ◽  
Author(s):  
Clemens Fuchs ◽  
Robert Tichy ◽  
Volker Ziegler

2008 ◽  
Vol 50 (1) ◽  
pp. 71-74
Author(s):  
NAHID ASHRAFI

AbstractThe unit sum number u(R) of a ring R is the least k such that every element is the sum of k units; if there is no such k then u(R) is ω or ∞ depending whether the units generate R additively or not. If RM is a left R-module, then the unit sum number of M is defined to be the unit sum number of the endomorphism ring of M. Here we show that if R is a ring such that R/J(R) is semisimple and $\Z_{2}$ is not a factor of R/J(R) and if P is a projective R-module such that JP ≪ P, (JP small in P), then u(P)= 2. As a result we can see that if P is a projective module over a perfect ring then u(P)=2.


2008 ◽  
Vol 133 (4) ◽  
pp. 297-308 ◽  
Author(s):  
Alan Filipin ◽  
Robert Tichy ◽  
Volker Ziegler

2007 ◽  
Vol 75 (3) ◽  
pp. 355-360 ◽  
Author(s):  
Dinesh Khurana ◽  
Ashish K. Srivastava

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.


2005 ◽  
Vol 56 (1) ◽  
pp. 1-12 ◽  
Author(s):  
N. Ashrafi
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