Euclidean Domains and Principal Ideal Domains

2018 ◽  
pp. 100-109
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Domains ◽  
Euclidean Domains

PRINCIPAL IDEAL DOMAINS AND EUCLIDEAN DOMAINS HAVING 1 AS THE ONLY UNIT

2001 ◽  
Vol 29 (11) ◽  
pp. 5197-5208
Author(s):  
William Heinzer ◽  
Moshe Roitman1*
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Domains ◽  
Euclidean Domains

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.

Right Invariant Right Hereditary Rings

1974 ◽  
Vol 26 (5) ◽  
pp. 1186-1191 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Maximal Orders ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Left Order ◽  
Principal Ideal Domains

Let R be a right hereditary domain in which all right ideals are two-sided (i.e., R is right invariant). We show that R is the intersection of generalized discrete valuation rings and that every right ideal is the product of prime ideals. This class of rings seems comparable with (and contains) the class of commutative Dedekind domains, but the rings considered here are in general not maximal orders and not Dedekind rings in the terminology of Robson [9]. The left order of a right ideal of such a ring is a ring of the same kind and the class contains right principal ideal domains in which the maximal right ideals are two-sided [6].

Radicals of PID's and Dedekind Domains

1972 ◽  
Vol 24 (4) ◽  
pp. 566-572 ◽  
Author(s):  
R. E. Propes
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Radical Class ◽  
Dedekind Domains ◽  
Radical Ideals ◽  
The One ◽  
Image Position ◽  
Principal Ideal Domains

The purpose of this paper is to characterize the radical ideals of principal ideal domains and Dedekind domains. We show that if T is a radical class and R is a PID, then T(R) is an intersection of prime ideals of R. More specifically, ifthen T(R) = (p1p2 … pk), where p1, p2, … , pk are distinct primes, and where (p1p2 … Pk) denotes the principal ideal of R generated by p1p2 … pk. We also characterize the radical ideals of commutative principal ideal rings. For radical ideals of Dedekind domains we obtain a characterization similar to the one given for PID's.

Sign in / Sign up

Export Citation Format

Share Document