scholarly journals Noetherian rings of injective dimension one which are orders in quasi-Frobenius rings

2003 ◽  
Vol 270 (1) ◽  
pp. 249-260 ◽  
Author(s):  
A.W Chatters ◽  
C.R Hajarnavis
Keyword(s):  
Noetherian Rings ◽  
Injective Dimension ◽  
Frobenius Rings ◽  
Dimension One

scholarly journals Some properties of non-commutative regular graded rings

1992 ◽  
Vol 34 (3) ◽  
pp. 277-300 ◽  
Author(s):  
Thierry Levasseur
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Supplementary Condition ◽  
Graded Rings ◽  
Finitely Generated ◽  
Injective Dimension ◽  
Commutative Case ◽  
Finite Global Dimension ◽  
Image Position ◽  
Dimension One

Let A be a noetherian ring. When A is commutative (of finite Krull dimension), A is said to be Gorenstein if its injective dimension is finite. If A has finite global dimension, one says that A is regular. If A is arbitrary, these hypotheses are not sufficient to obtain similar results to those of the commutative case. To remedy this problem, M. Auslander has introduced a supplementary condition. Before stating it, we recall that the grade of a finitely generated (left or right) module is defined by

Idealiser Rings of Injective Dimension One

2006 ◽  
Vol 13 (02) ◽  
pp. 239-252 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Injective Dimension ◽  
Projective Ideal ◽  
Dimension One

We study a certain type of prime Noetherian idealiser ring R of injective dimension 1, and prove for instance that the idempotent ideals of R are projective and that every non-zero projective ideal of R is uniquely of the form UE for some invertible ideal U and idempotent ideal E of R. Formulae are given for the number of idempotent ideals of R and the number of orders which contain R.

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