These two statements require some elaboration: first, commutative algebra, as a mainstay of the algebra part of a solid PhD program, is treated wonderfully well in a number of texts, and many instructors would opt for other sources. For one thing, the chapters in Atiyah-MacDonald are cut to the bone: although the discussions overflow with elegance and abound with precision, they are not easy to use as scripts for accessible lectures. The lecturer would have to do a lot of work in order properly to motivate his presentations, at least for an average class. However, if the onus is placed on the student, i. And this brings me to my second point: Atiyah-MacDonald is indeed unsurpassed as an exemplar of a text which has its pedagogical gravity located in the sets of exercises. The salient point is that the behavior of algebraic varieties, with the business of prime, primary, maximal and radical ideals, indeed, the back-and-forth between ideal theory and the inner life of zero sets of polynomials with the Zariski topology knocking at the door , is the heart of the subject immediately out of the gate, and a solid grounding in this aspect of commutative algebra is indispensable.

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Overview[ edit ] Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings , which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the notion of a valuation ring.

The restriction of algebraic field extensions to subrings has led to the notions of integral extensions and integrally closed domains as well as the notion of ramification of an extension of valuation rings.

The notion of localization of a ring in particular the localization with respect to a prime ideal , the localization consisting in inverting a single element and the total quotient ring is one of the main differences between commutative algebra and the theory of non-commutative rings. It leads to an important class of commutative rings, the local rings that have only one maximal ideal.

The set of the prime ideals of a commutative ring is naturally equipped with a topology , the Zariski topology.

All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of scheme theory , a generalization of algebraic geometry introduced by Grothendieck. Many other notions of commutative algebra are counterparts of geometrical notions occurring in algebraic geometry.

This is the case of Krull dimension , primary decomposition , regular rings , Cohenâ€”Macaulay rings , Gorenstein rings and many other notions. 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 , who recast many earlier results in terms of an ascending chain condition , now known as the Noetherian condition.

The main figure responsible for the birth of commutative algebra as a mature subject was Wolfgang Krull , who introduced the fundamental notions of localization and completion of a ring, as well as that of regular local rings. He established the concept of the Krull dimension of a ring, first for Noetherian rings before moving on to expand his theory to cover general valuation rings and Krull rings. These results paved the way for the introduction of commutative algebra into algebraic geometry, an idea which would revolutionize the latter subject.

Much of the modern development of commutative algebra emphasizes modules. Both ideals of a ring R and R-algebras are special cases of R-modules, so module theory encompasses both ideal theory and the theory of ring extensions. Main tools and results[ edit ] Main article: Noetherian ring In mathematics , more specifically in the area of modern algebra known as ring theory , a Noetherian ring, named after Emmy Noether , is a ring in which every non-empty set of ideals has a maximal element. Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals; that is, given any chain: I.

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Overview[ edit ] Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings , which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the notion of a valuation ring. The restriction of algebraic field extensions to subrings has led to the notions of integral extensions and integrally closed domains as well as the notion of ramification of an extension of valuation rings. The notion of localization of a ring in particular the localization with respect to a prime ideal , the localization consisting in inverting a single element and the total quotient ring is one of the main differences between commutative algebra and the theory of non-commutative rings.

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