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In other words, K[X] has the following universal property: For every ring R containing K, and every element a of R, there is a unique algebra homomorphism from K[X] to R that fixes K, and maps X to a. As for all universal properties, this defines the pair (K[X], X) up to a unique isomorphism, and can therefore be taken as a definition of K[X].
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them.
There is a (non-obvious) injective ring homomorphism from the ring of symmetric polynomials in variables to the ring of symmetric polynomials in + variables. Forming the direct limit of this direct system yields the ring of symmetric functions. Let F be a C-valued sheaf on a topological space X. Fix a point x in X.
The localization of a commutative ring R by a multiplicatively closed set S is a new ring whose elements are fractions with numerators in R and denominators in S.. If the ring is an integral domain the construction generalizes and follows closely that of the field of fractions, and, in particular, that of the rational numbers as the field of fractions of the integers.
The Weyl algebras are Ore extensions, with R any commutative polynomial ring, σ the identity ring endomorphism, and δ the polynomial derivative. Ore algebras are a class of iterated Ore extensions under suitable constraints that permit to develop a noncommutative extension of the theory of Gröbner bases.
The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. [11] In 1871, Richard Dedekind defined the concept of the ring of integers of a number field. [12] In this context, he introduced the terms "ideal" (inspired by Ernst Kummer's notion of ideal number) and "module" and studied their properties ...
In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λ n on it that behave like the exterior powers of vector spaces.Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results ...
For any local ring A there is a universal Henselian ring B generated by A, called the Henselization of A, introduced by Nagata (1953), such that any local homomorphism from A to a Henselian ring can be extended uniquely to B. The Henselization of A is unique up to unique isomorphism.