Commutative algebra

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In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules based on such rings; and of fields and their algebras. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers Z, and p-adic integers.

Given the scheme concept, much of commutative algebra can be seen as either the local or the affine theory of algebraic geometry.

The study of noncommutative rings is known as noncommutative algebra, consisting of ring theory, representation theory, and Banach algebra.

[edit] History

The subject, first known as ideal theory, began with Richard Dedekind's work on ideals based on the earlier work of Ernst Kummer and Leopold Kronecker. Later, David Hilbert introduced the term "ring" to generalize the earlier term "number ring". Hilbert introduced a more abstract approach to replace the more concrete and computationally oriented methods grounded in such things as complex analysis and classical invariant theory. In turn, Hilbert strongly influenced Emmy Noether, to whom we owe much of the abstract and axiomatic approach to the subject. Another important milestone was the work of Hilbert's student Emanuel Lasker (also a world chess champion), who introduced primary ideals and proved the first version of the Lasker–Noether theorem.

More recent work in commutative algebra emphasizes modules, a tendency, apparent in Kronecker, which blossomed in Noether's work. Modules constitute a technical improvement, in that they focus on ideals. An ideal a in a ring R and its quotient ring R/a can all be put on an equal footing as instances of modules.