scholarly journals Resolving Zero Divisors Using Hensel Lifting

2017 ◽  
Author(s):  
John Kluesner ◽  
Michael Monagan
Keyword(s):  
Hensel Lifting ◽  
Zero Divisors

Zero-Divisors, Torsion Elements, and Unions of Annihilators

2014 ◽  
Vol 43 (1) ◽  
pp. 76-83 ◽  
Author(s):  
D. D. Anderson ◽  
Sangmin Chun
Keyword(s):  
Zero Divisors

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

Rings with few zero divisors

1969 ◽  
Vol 181 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Basil Corbas
Keyword(s):  
Zero Divisors

On the Variety of Zero Divisors in Algebras

2020 ◽  
pp. 57-68
Author(s):  
Fabrizio Colombo ◽  
Irene Sabadini ◽  
Daniele C. Struppa
Keyword(s):  
Zero Divisors

scholarly journals Some noteworthy examples of zero divisors in infinite rings II

1998 ◽  
Vol 83 (2) ◽  
pp. 161
Author(s):  
Howard E. Bell ◽  
Abraham A. Klein
Keyword(s):  
Zero Divisors
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