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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 particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition , below).
However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem). Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.
Every vector space is a free module, [1] but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S.
In other words, the perfection R(A) of A is a perfect ring of characteristic p together with a map θ : R(A) → A such that for any perfect ring B of characteristic p equipped with a map φ : B → A, there is a unique map f : B → R(A) such that φ factors through θ (i.e. φ = θf). The perfection of A may be constructed as follows.
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.
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 .
Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is a ...