Search results
Results From The WOW.Com Content Network
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).
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.
If R = Π i∈I R i is a product of rings, then for every i in I we have a surjective ring homomorphism p i : R → R i which projects the product on the i th coordinate. The product R together with the projections p i has the following universal property:
For example, choosing a basis, a symmetric algebra satisfies the universal property and so is a polynomial ring. To give an example, let S be the ring of all functions from R to itself; the addition and the multiplication are those of functions. Let x be the identity function.
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.
As the latter formulation makes clear, a polynomial ring is the group algebra over a vector space, and writing in terms of corresponds to choosing a basis for the vector space. Then an ideal I, or equivalently a module R / I , {\displaystyle R/I,} is a cyclic representation of R (cyclic meaning generated by 1 element as an R -module; this ...
The same is true for a directed collection of subgroups of a given group, or a directed collection of subrings of a given ring, etc. The weak topology of a CW complex is defined as a direct limit. Let X {\displaystyle X} be any directed set with a greatest element m {\displaystyle m} .