ring hierarchy

Figure 1: Diagram of the hierarchy of rings.

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

•
Every ring considered here has a 1.
•
When one class of rings is connected to another class by a line, then the lower class is a subclass of the higher placed class.
•
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.
•
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.
•
However some subclasses are the result of deep theorems. For example, every artinian ring is also noetherian.

List of common rings

1.
Ring.
2.
Commutative ring.
3.
Noetherian ring (http://planetmath.org/Noetherian).
4.
Jacobson semisimple ring.
5.
Local ring.
6.
Integral domain.
7.
Artinian ring (http://planetmath.org/Artinian).
8.
Primitive ring.
9.
Unique factorization domain (UFD).
10.
Dedekind domain.
11.
Semisimple ring.
12.
Principal ideal domain (PID) (http://planetmath.org/PrincipalIdealDomain).
13.
Simple ring.
14.
Discrete valuation domain (DVD) (http://planetmath.org/DiscreteValuationRing) (Alsocalled a Discrete valuation ring).
15.
Euclidean domain.
16.
Division ring.
17.
Field.

The following containments are definitional:

•
Ring $>$ commutative ring, noetherian ring and Jacobson semisimple ring.
•
Commutative ring $>$ local ring and integral domain.
•
Integral domain $>$ unique factorization domain and Dedekind domain.
•
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 [Hopkins-Levitzki] [2, Theorem 8.46].
3.
Noetherian rings $>$ Dedekind domain [1, Theorem VIII.6.10].
4.
Artinian rings $>$ semisimple rings, [Wedderburn-Artin theorem].¹¹Some definitions semisimple make this containment part of the definition. Otherwise the result is part of the Wedderburn-Artin theorem.
5.
Jacobson semisimple $>$ semisimple rings.[Wedderburn-Artin theorem].²²Also 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, 73Springer-Verlag, 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.

单词	RingHierarchy
释义	ring hierarchy Figure 1: Diagram of the hierarchy of rings. The objects in the diagram reflect many of the common rings encountered in ring theory. • Every ring considered here has a 1. • When one class of rings is connected to another class by a line, then the lower class is a subclass of the higher placed class. • 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. • 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. • However some subclasses are the result of deep theorems. For example, every artinian ring is also noetherian. List of common rings 1. Ring. 2. Commutative ring. 3. Noetherian ring (http://planetmath.org/Noetherian). 4. Jacobson semisimple ring. 5. Local ring. 6. Integral domain. 7. Artinian ring (http://planetmath.org/Artinian). 8. Primitive ring. 9. Unique factorization domain (UFD). 10. Dedekind domain. 11. Semisimple ring. 12. Principal ideal domain (PID) (http://planetmath.org/PrincipalIdealDomain). 13. Simple ring. 14. Discrete valuation domain (DVD) (http://planetmath.org/DiscreteValuationRing) (Alsocalled a Discrete valuation ring). 15. Euclidean domain. 16. Division ring. 17. Field. The following containments are definitional: • Ring $>$ commutative ring, noetherian ring and Jacobson semisimple ring. • Commutative ring $>$ local ring and integral domain. • Integral domain $>$ unique factorization domain and Dedekind domain. • 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 [Hopkins-Levitzki] [2, Theorem 8.46]. 3. Noetherian rings $>$ Dedekind domain [1, Theorem VIII.6.10]. 4. Artinian rings $>$ semisimple rings, [Wedderburn-Artin theorem].¹¹Some definitions semisimple make this containment part of the definition. Otherwise the result is part of the Wedderburn-Artin theorem. 5. Jacobson semisimple $>$ semisimple rings.[Wedderburn-Artin theorem].²²Also 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, 73Springer-Verlag, 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.
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