ring hierarchy
The objects in the diagram reflect many of the common rings encountered in ring theory.

•
Every ring considered here has a 1.
 •

•
If a class has more than one parent in the graph it is not always the case that this class represents the strict intersection^{} of these two classes, but it is certainly contained in this intersection.

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Many of these containments are trivial in the sense that they are defined as subclasses of one another. For instance, principal ideal domain^{} is by definition a domain.

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However some subclasses are the result of deep theorems. For example, every artinian ring is also noetherian^{}.
List of common rings

1.
Ring.
 2.

3.
Noetherian ring (http://planetmath.org/Noetherian).
 4.

5.
Local ring^{}.

6.
Integral domain^{}.

7.
Artinian ring (http://planetmath.org/Artinian^{}).
 8.

9.
Unique factorization domain^{} (UFD).

10.
Dedekind domain^{}.
 11.

12.
Principal ideal domain (PID) (http://planetmath.org/PrincipalIdealDomain).

13.
Simple ring^{}.

14.
Discrete valuation domain (DVD) (http://planetmath.org/DiscreteValuationRing) (Also called a Discrete valuation ring).

15.
Euclidean domain^{}.
 16.

17.
Field.
The following containments are definitional:

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Ring $>$ commutative ring, noetherian ring and Jacobson semisimple ring.

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Commutative ring $>$ local ring and integral domain.

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Integral domain $>$ unique factorization domain and Dedekind domain.

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Semisimple rings $>$ simple rings.

•
Local rings $>$ Discrete valuation domains.

•
Principal ideal domains $>$ Discrete valuation domains.

•
Division rings $>$ fields.
The following containments are due to theorems:

1.
Jacobson semisimple rings $>$ primitive rings [2, p. 571].

2.
Noetherian rings $>$ artinian rings [HopkinsLevitzki] [2, Theorem 8.46].

3.
Noetherian rings $>$ Dedekind domain [1, Theorem VIII.6.10].

4.
Artinian rings $>$ semisimple rings, [WedderburnArtin theorem]. ^{1}^{1}Some definitions semisimple^{} make this containment part of the definition. Otherwise the result is part of the WedderburnArtin theorem.

5.
Jacobson semisimple $>$ semisimple rings.[WedderburnArtin theorem].^{2}^{2}Also depends on the definition of semisimple.

6.
Dedekind domain $>$ Principal ideal domain [1, p. 401].

7.
Principal ideal domains $>$ euclidean domains [2, Theorem 3.60].

8.
Simple rings $>$ division rings.
References
 1 Hungerford, Thomas W. Algebra, Graduate Texts in Mathematics, 73 SpringerVerlag, New York, (1980), pp. xxiii+502.
 2 Rotman, Joseph J. Advanced modern algebra, Prentice Hall Inc.,Upper Saddle River, NJ, (2002), pp xvi+1012+A8+B6+I14.
Title  ring hierarchy 

Canonical name  RingHierarchy 
Date of creation  20130322 16:01:16 
Last modified on  20130322 16:01:16 
Owner  Algeboy (12884) 
Last modified by  Algeboy (12884) 
Numerical id  8 
Author  Algeboy (12884) 
Entry type  Topic 
Classification  msc 06E20 