All vector spaces are by default over or . We consider three vector spaces , and . Let be a tensor product of and , and a tensor product of and . The aim is first to show that , together with the mapping: is a tensor product of , and . Proof:…
Vector spaces and associative algebras will be implicitely all on the same field or . Let be a vector space. We wish to build a free associative algebra on . For any natural number , we will note (tensor product times). Each is a vector space. We may form: An element of can…
We assume that is an infinite set. Cofinite filters The cofinite filter over is the set of subsets of such that is finite. Since is infinite, is not in . It is trivial to check that any superset in of an element of is in , and that the intersection of any two elements of…
Two cases, that I have been unable to unify: a filter base over a power set with a set, or over . First case: filter bases over a power set filter base over , with a set. Definitions Property It is always the case that . Proof: Let be an eventual…
Let and be filters on a same set . Then is also a filter on . Proof: belongs to both and , hence it belongs to , and . Since is not an element of , it is not an element of . Let be an element of and a superset of . Then and…
A category is by definition concrete if there exists a faithful functor from to the category of sets . (A functor is faithful if it is one-to-one (injective) regarding morphisms.) It happens to be the case that in the category , monomorphisms and one-to-one mappings are the same thing; similarly for epimorphisms and surjections. We…
The theorem If and are such that is separately coprime with each , then is coprime with their product . The proof Proof: Through Bézout. being coprime with each , we can write, for each : If we take the product of each of these expressions, we get: Among the terms of this product, all…
For the definition and characterization of the polynomials over a field , see Polynomials over a field. Definition of the degree of a polynomial The degree of a polynomial P, written , is an element of . The symbol is taken to have the usual properties of order and of addition relative to natural numbers….
In this context, we consider a commutative field (simply: field) . The polynomials will be constructed as a certain associative unital algebra over (“-aua”), together with a distinguished element called the formal variable. itself, as a vector space over itself and the usual multiplication in as the third law, is a -aua, Morphisms between two…
Definition Let be a commutative field. The polynomials form a commutative unital algebra over containing an element such that the following universal property holds: For any unital algebra (commutative or otherwise) and any element , there exists exactly one unital algebra morphism such that . If and are elements of , let be the unique…