Principal ideal ring

From Wikipedia, the free encyclopedia

In mathematics, a principal ideal ring, or simply principal ring, is a commutative ring R such that every ideal I of R is a principal ideal, i.e. generated by a single element a of R.

A principal ideal ring which is also an integral domain is said to be a principal ideal domain (PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain.

[edit] Examples

1. Let $R_1,\ldots,R_n$ be rings and $R = \prod_{i=1}^n R_i$ . Then R is a principal ring if and only if R_i is a principal ring for all i.

2. The localization of a principal ring at any multiplicative subset is again a principal ring. Similarly, any quotient of a principal ring is again a principal ring.

3. Let R be a Dedekind domain and I be a nonzero ideal of R. Then the quotient R/I is a principal ring. Indeed, we may factor I as a product of prime powers: $I = \prod_{i=1}^n P_i^{a_i}$ , and by the Chinese Remainder Theorem $R/I \cong \prod_{i=1}^n R/P_i^{a_i}$ , so it suffices to see that each $R/P_i^{a_i}$ is a principal ring. But $R/P_i^{a_i}$ is isomorphic to the quotient $R_{P_i}/P_i^{a_i} R_{P_i}$ of the discrete valuation ring $R_{P_i}$ and, being a quotient of a principal ring, is itself a principal ring.

4. Let k be a finite field and put $A = k [x, y]$ , $\mathfrak{m} = \langle x, y \rangle$ and $R = A/\mathfrak{m}^2$ . Then R is a finite local ring which is not principal.

[edit] Structure theory

The principal rings constructed in Example 3. above are always Artinian rings; in particular they are isomorphic to a finite direct product of principal Artinian local rings. A local Artinian principal ring is called a special principal ring and has an extremely simple ideal structure: there are only finitely many ideals, each of which is a power of the maximal ideal.

The following result gives a complete classification of principal rings in terms of special principal rings and principal ideal domains.

Theorem (Zariski-Samuel): Let R be a principal ring. Then R can be written as a direct product $\prod_{i=1}^n R_i$ , where each R_i is either a principal ideal domain or a special principal ring.

The proof applies the Chinese Remainder theorem to a minimal primary decomposition of the zero ideal.

There is also the following result, due to Hungerford:

Theorem (Hungerford): Let R be a principal ring. Then R can be written as a direct product $\prod_{i=1}^n R_i$ , where each R_i is a quotient of a principal ideal domain.

The proof of Hungerford's theorem employs Cohen's structure theorems for complete local rings.

Arguing as in Example 3. above and using the Zariski-Samuel theorem, it is easy to check that Hungerford's theorem is equivalent to the statement that any special principal ring is the quotient of a discrete valuation ring.

[edit] References

T. Hungerford, On the structure of principal ideal rings, Pacific J. Math. 25 1968 543--547.

Pages 86 & 146-155 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0

Zariski, O.; Samuel, P. (1975), Commutative algebra, Graduate Texts in Mathematics, 28, 29, Berlin, New York: Springer-Verlag

Principal ideal ring

From Wikipedia, the free encyclopedia

[edit] Examples

[edit] Structure theory

[edit] References

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