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Formally, a ring is a set endowed with two binary operations called addition and multiplication such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element.
In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality is always true in elementary algebra. For example, in elementary arithmetic, one has Therefore, one would say that multiplication distributes over addition. This basic property of numbers is part of the ...
By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a rng. For a rng R, a left ideal I is a subrng with the additional property that is in I for every and every . (Right and two-sided ideals are defined similarly.)
A ring inherits some "good" properties from its associated graded ring. For example, if R is a noetherian local ring, and is an integral domain, then R is itself an integral domain. gr of a quotient module. Let be left modules over a ring R and I an ideal of R. Since
The category Ring is a concrete category meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure. There is a natural forgetful functor. U : Ring → Set. for the category of rings to the category of sets which sends each ring to its underlying set ...
t. e. In algebra, ring theory is the study of rings [1] — algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings ...
The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from the relation X⋅r = f(r)⋅X to produce an associative multiplication that distributes over the standard addition.
Divisibility (ring theory) In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. Divisibility is a useful concept for the analysis of the structure of ...