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Commutative Algebra

Let A denote an R-algebra, so that A is a vector space over R and

A×A->A
(1)
(x,y)|->x·y.
(2)

Now define

Z={x in A:x·y=0 for some y in A!=0},
(3)

where 0 in Z. An Associative R-algebra is commutative if x·y=y·x for all x,y in A. Similarly, a ring is commutative if the multiplication operation is commutative, and a Lie algebra is commutative if the commutator [A,B] is 0 for every A and B in the Lie algebra.

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." http://algo.inria.fr/csolve/zerodiv/.Kreuzer, M. and Robbiano, L. Computational Commutative Algebra 1. Berlin: 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.Zariski, O. and Samuel, P. Commutative Algebra I. New York: Springer-Verlag, 1958.

Referenced on Wolfram|Alpha

Commutative Algebra

Cite this as:

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

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