Algebra (ring theory) Ring (mathematics) Chapter IX of David Hilbert's Die Theorie der algebraischen Zahlkörper. The chapter title is Die Zahlringe des Körpers, literally "the number rings of the field". The word "ring" is the contraction of "Zahlring". Rings were first formalized as a common generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory.
They are also used in other branches of mathematics such as geometry and mathematical analysis. Whether a ring is commutative or not has profound implication in the study of rings as abstract objects, the field called the ring theory. The most familiar example of a ring is the set of all integers, Z, consisting of the numbers The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings.
R is an abelian group under addition, meaning: 1. 2. 3. 4. a + b = b + a for all a and b in R (+ is commutative). 5. 6. 8. Equip the set in Z4 is . Ring theory. Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to. For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra.
Similarly, Fermat's last theorem is stated in terms of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry. History[edit] Commutative rings[edit] Algebraic geometry[edit] Noncommutative rings[edit] Ideal (ring theory) In ring theory , a branch of abstract algebra , an ideal is a special subset of a ring . Ideals generalize certain subsets of the integers , such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal.
Among the integers, the ideals correspond one-for-one with the non-negative integers : in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may be distinct from the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. The concept of an order ideal in order theory is derived from the notion of ideal in ring theory. History [ edit ] Definitions [ edit ] For an arbitrary ring , let is a subgroup of of and. Ringtheorie. Die Ringtheorie ist ein Teilgebiet der Algebra , das sich mit den Eigenschaften von Ringen beschäftigt. Ein Ring ist eine algebraische Struktur , in der, ähnlich wie in den ganzen Zahlen , Addition und Multiplikation definiert und miteinander bezüglich Klammersetzung verträglich sind. Namensgebung [ Bearbeiten ] Der Name Ring bezieht sich nicht auf etwas anschaulich Ringförmiges, sondern auf einen organisierten Zusammenschluss von Elementen zu einem Ganzen.
Diese Wortbedeutung ist in der deutschen Sprache ansonsten weitgehend verloren gegangen. Einige ältere Vereinsbezeichnungen (wie z. Definitionen [ Bearbeiten ] Ring [ Bearbeiten ] Ein Ring ist eine Menge mit zwei inneren binären Verknüpfungen und , sodass für alle erfüllt sind. Das neutrale Element von heißt Nullelement des Rings . Zusätzlich ein neutrales Element , ist also ein Monoid , dann nennt man einen Ring mit Eins oder unitären Ring . Ist die Halbgruppe Abschwächung der Axiome [ Bearbeiten ] gilt: Addiert man diese Gleichung von links mit , wenn.