Principal ideal theorems

1979 ◽  
pp. 174-206
Author(s):  
Melvin Hochster
Keyword(s):  
Principal Ideal

Eigenvalues of zero divisor graphs of principal ideal rings

2021 ◽  
pp. 1-15
Author(s):  
Jitsupat Rattanakangwanwong ◽  
Yotsanan Meemark
Keyword(s):  
Principal Ideal ◽  
Zero Divisor

scholarly journals Torsion and protorsion modules over free ideal rings

1970 ◽  
Vol 11 (4) ◽  
pp. 490-498
Author(s):  
P. M. Cohn
Keyword(s):  
Principal Ideal ◽  
Finitely Generated ◽  
Commutative Case ◽  
Principal Ideal Domains ◽  
Torsion Modules ◽  
Noncommutative Analogue

Free ideal rings (or firs, cf. [2, 3] and § 2 below) form a noncommutative analogue of principal ideal domains, to which they reduce in the commutative case, and in [3] a category TR of right R-modules was defined, over any fir R, which forms an analogue of finitely generated torsion modules. The category TR was shown to be abelian, and all its objects have finite composition length; more over, the corresponding category RT of left R-modules is dual to TR.

scholarly journals SUBALGEBRAS OF THE POLYNOMIAL ALGEBRA IN POSITIVE CHARACTERISTIC AND THE JACOBIAN

2011 ◽  
Vol 10 (04) ◽  
pp. 605-613
Author(s):  
ALEXEY V. GAVRILOV
Keyword(s):  
Principal Ideal ◽  
Positive Characteristic ◽  
Polynomial Algebra ◽  
Characteristic P ◽  
The Ideal

Let 𝕜 be a field of characteristic p > 0 and R be a subalgebra of 𝕜[X] = 𝕜[x1, …, xn]. Let J(R) be the ideal in 𝕜[X] defined by [Formula: see text]. It is shown that if it is a principal ideal then [Formula: see text], where q = pn(p - 1)/2.

scholarly journals A note on the invertible ideal theorem

1984 ◽  
Vol 25 (1) ◽  
pp. 27-30 ◽  
Author(s):  
Andy J. Gray
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Simple Proof ◽  
Noetherian Ring ◽  
Principal Ideal ◽  
Central Element ◽  
Commutative Case ◽  
Additional Information ◽  
The Subject

This note is devoted to giving a conceptually simple proof of the Invertible Ideal Theorem [1, Theorem 4·6], namely that a prime ideal of a right Noetherian ring R minimal over an invertible ideal has rank at most one. In the commutative case this result may be easily deduced from the Principal Ideal Theorem by localizing and observing that an invertible ideal of a local ring is principal. Our proof is partially analogous in that it utilizes the Rees ring (denned below) in order to reduce the theorem to the case of a prime ideal minimal over an ideal generated by a single central element, which can be easily dealt with by adapting the commutative argument in [8]. The reader is also referred to the papers of Jategaonkar on the subject [5, 6, 7], particularly the last where another proof of the theorem appears which yields some additional information.

scholarly journals Intersections of powers of a principal ideal and primality

2004 ◽  
Vol 188 (1-3) ◽  
pp. 287-304
Author(s):  
Paolo Zanardo
Keyword(s):  
Principal Ideal

scholarly journals Notes on topological rings

2013 ◽  
Vol 29 (2) ◽  
pp. 267-273
Author(s):  
MIHAIL URSUL ◽  
◽  
MARTIN JURAS ◽  
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Domain ◽  
Compact Ring ◽  
Ring Topology ◽  
Bezout Domain ◽  
Totally Bounded ◽  
Ideal Domain ◽  
Nilpotent Ring ◽  
Trivial Multiplication

We prove that every infinite nilpotent ring R admits a ring topology T for which (R, T ) has an open totally bounded countable subring with trivial multiplication. A new example of a compact ring R for which R2 is not closed, is given. We prove that every compact Bezout domain is a principal ideal domain.

On (α + uβ)-constacyclic codes of length 4ps over 𝔽pm + u𝔽pm∗

2019 ◽  
Vol 18 (02) ◽  
pp. 1950023 ◽  
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta ◽  
Thang M. Vo
Keyword(s):  
Principal Ideal ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Chain Ring ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring ◽  
Unit Formula

For any odd prime [Formula: see text] such that [Formula: see text], the structures of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text] [Formula: see text] are established in term of their generator polynomials. When the unit [Formula: see text] is a square, each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of two constacyclic codes of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that the ambient ring [Formula: see text] is a principal ideal ring. From that, the structure, number of codewords, duals of all such [Formula: see text]-constacyclic codes are obtained. As an application, we identify all self-orthogonal, dual-containing, and the unique self-dual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text].

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