algebra (ring theory) - Article Index

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Motivation
In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Relation To Representation Theory
If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Types Of Modules
Finitely generated. A module M is finitely generated if there exist finitely many elements x1,...,xn in M such that every element of M is...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Generalizations
Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left R-module is nothing but a (covar...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Definition
Specifically, a left module over the ring R consists of an abelian group (M, +) and an operation R × M → M (called scalar multiplicati...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Submodules And Homomorphisms
Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or R-submodule, to be more explicit) if, for any n in N and...   » Read the article

Encyclopedia - Associative Algebra: Encyclopedia Ii - Associative Algebra - Definition
An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A → A (whe...   » Read the article

Encyclopedia - Associative Algebra: Encyclopedia Ii - Associative Algebra - Coalgebras
An associative unitary algebra over K is based on a morphism A×A→A having 2 inputs (multiplicator and multiplicand) and one output (pr...   » Read the article

Encyclopedia - Associative Algebra: Encyclopedia Ii - Associative Algebra - Generalizations
One may consider associative algebras over a commutative ring R: these are modules over R together with a R-bilinear map which yields an ...   » Read the article

Encyclopedia - Associative Algebra: Encyclopedia Ii - Associative Algebra - Algebra Homomorphisms
If A and B are associative algebras over the same field K, an algebra homomorphism h: A → B is a K-linear map which is also multiplicat...   » Read the article

Encyclopedia - Associative Algebra: Encyclopedia Ii - Associative Algebra - Index-free Notation
In the above definition of an associative algebra, the definition of associativity was made with regard to all of the elements of A. It i...   » Read the article