ring of principal ideals

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• Ascending chain condition on principal ideals — In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two sided ideals of a ring, partially ordered by inclusion. The ascending ascending chain condition on principal… …   Wikipedia

• Ring theory — In abstract algebra, ring theory is the study of rings algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. Ring theory studies the structure of rings, their… …   Wikipedia

• Principal ideal — In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R .More specifically: * a left principal ideal of R is a subset of R of the form R a := { r a : r in R }; * a… …   Wikipedia

• Ring (mathematics) — This article is about algebraic structures. For geometric rings, see Annulus (mathematics). For the set theory concept, see Ring of sets. Polynomials, represented here by curves, form a ring under addition and multiplication. In mathematics, a… …   Wikipedia

• Glossary of ring theory — Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. Contents 1 Definition of a ring 2 Types of… …   Wikipedia

• Principal ideal ring — In mathematics, a principal ideal ring, or simply principal ring, is a ring R such that every ideal I of R is a principal ideal, i.e. generated by a single element a of R .A principal ideal ring which is also an integral domain is said to be a… …   Wikipedia

• Principal ideal theorem — This article is about the Hauptidealsatz of class field theory. You may be seeking Krull s principal ideal theorem, also known as Krull s Hauptidealsatz, in commutative algebra In mathematics, the principal ideal theorem of class field theory, a… …   Wikipedia

• Commutative ring — In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Some specific kinds of commutative rings are given with …   Wikipedia

• Ideal (ring theory) — In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like even number or multiple of 3 . For instance, in… …   Wikipedia

• Boolean ring — In mathematics, a Boolean ring R is a ring (with identity) for which x 2 = x for all x in R ; that is, R consists only of idempotent elements.Boolean rings are automatically commutative and of characteristic 2 (see below for proof). A Boolean… …   Wikipedia

• Divisibility (ring theory) — In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. Please see the page about divisors for this simplest example. With the development of abstract rings, of which the integers are the… …   Wikipedia