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Commutative Algebra

A commutative algebra over a field k is a vector space A over k equipped with bilinear multiplication satisfying ab=ba for all a,b in A. The usual associative algebra convention also requires multiplication to be associative and to have a multiplicative identity (Wisbauer 1991, p. 30). For example, the polynomial ring k[x] and the ring of dual numbers k[epsilon]/(epsilon^2) are commutative algebras over k.

More generally, for a commutative ring R with multiplicative identity, a unital commutative R-algebra is a commutative ring A with a specified identity-preserving ring homomorphism R->A. This ring homomorphism need not be injective. For instance, Z/2Z is a commutative Z-algebra via the map n|->n+2Z. For a Lie algebra, the term "commutative" conventionally means that the Lie bracket is zero for every pair of elements.

The term "commutative algebra" also refers to the branch of abstract algebra that studies commutative rings. Commutative algebra is important in algebraic geometry.

See also

Abelian Group, Abstract Algebra, Commutative, Commutative Ring

Portions of this entry contributed by John Renze

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References

Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, pp. 9-10, 1969.Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer-Verlag, 1996.Eisenbud, D. (Ed.). Commutative Algebra, Algebraic Geometry, and Computational Methods. Singapore: Springer-Verlag, 1999.Finch, S. "Zero Divisor Structure in Real Algebras." https://web.archive.org/web/20010418202727/http://www.mathsoft.com/asolve/zerodiv/zerodiv.html.Kreuzer, M. and Robbiano, L. Computational Commutative Algebra 1. Berlin, Germany: Springer-Verlag, 2000.MacDonald, I. G. and Atiyah, M. F. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969.Samuel, P. and Zariski, O. Commutative Algebra II. New York: Springer-Verlag, 1997.Wisbauer, R. Foundations of Module and Ring Theory: A Handbook for Study and Research. Reading: Gordon and Breach, 1991. https://www.math.uni-duesseldorf.de/~wisbauer/book.pdf.Zariski, O. and Samuel, P. Commutative Algebra I. New York: Springer-Verlag, 1958.

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Commutative Algebra

Cite this as:

Weisstein, Eric W., with contributions by John Renze. "Commutative Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CommutativeAlgebra.html

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