13 - Commutative rings and algebras

Introduction

A commutative ring is a set on which both "addition" and "multiplication" are defined, subject to the commutative, associative, and distributive laws; there is an element "0" with respect to which every element has a negative. Most authors also assume there is an element "1" with 1*x=x for all x, but we certainly do _not_ assume all nonzero elements have inverses. An obvious, very important example of a commutative ring is the ring of integers. General commutative ring theory Ring extensions and related topics Theory of modules and ideals (Co)homological methods Chain conditions, finiteness conditions Arithmetic rings and other special rings, see also 12-XX Integral domains Local rings and semilocal rings Finite commutative rings, For number-theoretic aspects, see 11TXX Topological rings and modules, see also 16W60, 16W80 Witt vectors and related rings Applications of logic to commutative algebra, See Also 03CXX, 03HXX Differential algebra, see also 12H05, 14F10 Computational aspects of commutative algebra. See also 68Q40

History

Applications and related fields

The rings considered in this section are both commutative and associative. For rings which are not assumed to be commutative, see Associative Rings and Algebras. Rings (commutative or otherwise) which are not even assumed associative, such as Lie algebras, are part of Nonassociative Rings.

Commutative rings with the added feature that all nonzero elements have inverses are fields, the topic of 12: Field Theory and Polynomials.

Generally, the rings of integers in number fields are part of algebraic number theory, but I have included here some files regarding their ring structure (e.g. Euclidean domains and Unique Factorization Domains (UFDs).)

For studies of sets of ideals in rings (particularly rings of polynomials) see Algebraic Geometry. In some sense algebraic geometry and commutative ring theory are simply two languages for the same ideas.

One common form of commutative ring is the ring of polynomials over a field (say). The algebraic study of _general collections_ of polynomials is appropriate for this field; the study of _individual_ polynomials or _specific_ collections usually belongs elsewhere.

• One may seek the roots of a polynomial numerically in Numerical Analysis; one may view them as analytic functions in Real analysis or Complex analysis.
• Specific families of polynomials, e.g. the Tchebyshev polynomials, are treated in Special functions, or Number Theory or Fourier series (and orthogonal functions) as appropriate.
• The general algebraic inspection of an integral polynomial could appropriate occur in many subfields of number theory (11) or field theory (12). For example, one may look at the extension field it generates in algebraic number theory or as a part of Galois theory. In general, I advise starting on the latter page for this material.

I wrote a long ring FAQ, a particular emphasis of which is the study of division rings over the reals, but which also includes some topics in commutative ring theory.

For Boolean rings see Ordered algebraic structures.

Subfields

• .13A: General commutative ring theory
• .13B: Ring extensions and related topics
• .13C: Theory of modules and ideals
• .13D: (Co)homological methods
• .13E: Chain conditions, finiteness conditions
• .13F: Arithmetic rings and other special rings, see also 12-XX
• .13G: Integral domains
• .13H: Local rings and semilocal rings
• .13J: Topological rings and modules, see also 16W60, 16W80
• .13K: Witt vectors and related rings
• .13L: Applications of logic to commutative algebra, See Also 03CXX, 03HXX
• .13M: Finite commutative rings, For number-theoretic aspects, see 11TXX
• .13N: Differential algebra, see also 12H05, 14F10
• 13P: Computational aspects of commutative algebra. See also 68Q40

Textbooks, reference works, and tutorials

"Reviews in Ring Theory, 1980-1984", American Mathematical Society, 1986 ISBN 0-8218-0097-3. Also available: reviews 1940-1979.

Hutchins, Harry C.: "Examples of commutative rings", Polygonal Publ. House, Washington, N. J., 1981. 167 pp. ISBN 0-936428-05-8

Kaplansky, Irving: "Commutative rings", revised edition, The University of Chicago Press, Chicago, Ill.-London, 1974. 182pp.

An on-line textbook. [John Beachy]

Online notes for a graduate course in ring theory. [Lee Lady]

Bourbaki, Nicolas, "Commutative algebra" Chapters 1--7: Springer-Verlag, Berlin-New York, 1989. 625 pp. ISBN 3-540-19371-5 (MR90a:13001); Chapters 8-9 [French only]: Masson, Paris, 1983. 200 pp. ISBN 2-225-78716-6

Software and tables

GB, A package for doing computations with Groebner Bases.

Macaulay.

CoCoA -- Computations in Commutative Algebra.

Relevant files at this site

• What is the group of units in the ring Z/mZ ? (When is it cyclic?)
• One may ask about quadratic extensions of the integers -- which are Euclidean, factorial, or PIDs.
• Citation and pointer for survey of Euclidean number rings and recent results.
• What is a free module and what are some modules that aren't free?
• What is the ABC theorem for polynomial rings (or, the ABC conjecture for the ring of integers).
• Proofs of Fermat's Last Theorem for polynomial rings (using Mason's ABC theorem or Wronskians)
• What is an order (in ring theory)
• Proving Fermat's Last Theorem in rings Z[x,y] or C(x).
• Connections between Hensel's lemma and Newton's method (Lit review)

Other web sites with this focus

• AMS resource page for this area.

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