Some remarks on Prüfer rings with zero-divisors

2021 ◽  
Vol 225 (9) ◽  
pp. 106692
Author(s):  
Federico Campanini ◽  
Carmelo Antonio Finocchiaro
Keyword(s):  
Zero Divisors ◽  
Prüfer Rings

scholarly journals Commutative rings with comparable regular elements

1996 ◽  
Vol 60 (1) ◽  
pp. 90-108 ◽  
Author(s):  
Paolo Zanardo
Keyword(s):  
Valuation Ring ◽  
Commutative Rings ◽  
The Other ◽  
Regular Elements ◽  
Zero Divisors ◽  
Valuation Rings ◽  
A Chain ◽  
Prüfer Rings ◽  
The Ideal

AbstractLet ℜ be the class of commutative rings R with comparable regular elements, that is, given two non zero-divisors in R, one divides the other. Applying the notion of V-valuation due to Harrison and Vitulli, we define the class V-val of V-valuated rings, which is contained in ℜ and contains the class of Manis valuation rings. We prove that these inclusions of classes are both proper. We investigate Prüfer rings inside ℜ, showing that there exist Prüfer rings which lie in ℜ but not in V-val; we prove that a ring R is a Prüfer valuation ring if and only if it is Prüfer and V-valuated, if and only if its lattice of regular ideals is a chain. Finally, we introduce and investigate the ideal I∞ of a ring R ∈ ℜ, which corresponds to the counterimage of ∞, whenever R is V-valuated.

scholarly journals Prüfer rings

1973 ◽  
Vol 5 (2) ◽  
pp. 132-138
Author(s):  
Paul Camion ◽  
L.S. Levy ◽  
H.B. Mann
Keyword(s):  
Prüfer Rings

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.

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