# ring hierarchy

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

List of common rings

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. 1.

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

2. 2.

Noetherian rings $>$ artinian rings [Hopkins-Levitzki] [2, Theorem 8.46].

3. 3.

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

4. 4.
5. 5.

Jacobson semisimple $>$ semisimple rings.[Wedderburn-Artin theorem].22Also depends on the definition of semisimple.

6. 6.

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

7. 7.

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

8. 8.

Simple rings $>$ division rings.

## References

• 1 Hungerford, Thomas W. Algebra, Graduate Texts in Mathematics, 73 Springer-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.
Title ring hierarchy RingHierarchy 2013-03-22 16:01:16 2013-03-22 16:01:16 Algeboy (12884) Algeboy (12884) 8 Algeboy (12884) Topic msc 06E20