multiplicatively closed subset

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• Multiplicatively closed set — In abstract algebra, a subset of a ring is said to be multiplicatively closed if it is closed under multiplication (i.e., xy is in the set when x and y are in it) and contains 1 but doesn t contain 0.[1] The condition is especially important in… …   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

• Localization of a module — In mathematics, the localization of a module is a construction to introduce denominators in a module for a ring. More precisely, it is a systematic way to construct a new module S −1 M out of a given module M containing fractions… …   Wikipedia

• Integrality — In commutative algebra, the notions of an element integral over a ring (also called an algebraic integer over the ring), and of an integral extension of rings, are a generalization of the notions in field theory of an element being algebraic over …   Wikipedia

• Torsion (algebra) — In abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be free. Definition Let G be a group. An element g of G is called a torsion element if g has… …   Wikipedia

• Boolean prime ideal theorem — In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstract algebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime… …   Wikipedia

• 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

• 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

• Flat module — In abstract algebra, a flat module over a ring R is an R module M such that taking the tensor product over R with M preserves exact sequences.Vector spaces over a field are flat modules. Free modules, or more generally projective modules, are… …   Wikipedia

• Semi-local ring — In mathematics, a semi local ring is a ring with a finite number of maximal ideals. It must be said that some refer to such a ring in general as a quasi local ring, using semi local ring to refer to a Noetherian ring with finitely many maximal… …   Wikipedia

• Overring — In mathematics, an overring B of an integral domain A is a subring of the field of fractions K of A that contains A: i.e., . A typical example is given by localization: if S is a multiplicatively closed subset of A, then the localization S−1A is… …   Wikipedia