Monoid ring
Encyclopedia
In abstract algebra
, a monoid ring is a new ring constructed from some other ring and a monoid
.
by (φ * ψ)(g) = Σkl=gφ(k)ψ(l).
The set of all such functions φ, together with these two operations, forms a ring, the monoid ring of G over R denoted R[G]. If G is a group
, then R[G] denotes the group ring
of G over R.
Less rigorously but more simply, an element of R[G] is a polynomial
in G over R, hence the notation. We multiply elements as polynomial
s, taking the product in G of the "indeterminates" and gathering terms:
where risj is the R-product and gihj is the G-product.
The ring R can be embedded in the ring R[G] via the ring homomorphism
T : R → R[G] defined by
where 1G is the identity element
of G.
There also exists a canonical homomorphism
going the other way, called the augmentation. It is the map ηR:R[G] → R ,defined by
The kernel of this homomorphism, the augmentation ideal, is denoted by JR(G). It is a free
R-module
generated by the elements 1 - g, for g in G.
s N (or {xn} viewed multiplicatively), we obtain the ring R[{xn}] =: R[x] of polynomial
s over R.
The Monoid Nn (with the addition) gives the polynomial ring with n variables: R[Nn] =: R[X1, ..., Xn].
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a monoid ring is a new ring constructed from some other ring and a monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
.
Definition
Let R be a ring and G be a monoid. Consider all the functions φ : G → R such that the set {g: φ(g) ≠ 0} is finite. Let all such functions be element-wise addable. We can define multiplicationby (φ * ψ)(g) = Σkl=gφ(k)ψ(l).
The set of all such functions φ, together with these two operations, forms a ring, the monoid ring of G over R denoted R[G]. If G is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, then R[G] denotes the group ring
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
of G over R.
Less rigorously but more simply, an element of R[G] is a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
in G over R, hence the notation. We multiply elements as polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s, taking the product in G of the "indeterminates" and gathering terms:
where risj is the R-product and gihj is the G-product.
The ring R can be embedded in the ring R[G] via the ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
T : R → R[G] defined by
- T(r)(1G) = r, T(r)(g) = 0 for g ≠ 1G.
where 1G is the identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
of G.
There also exists a canonical homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
going the other way, called the augmentation. It is the map ηR:R[G] → R ,defined by
The kernel of this homomorphism, the augmentation ideal, is denoted by JR(G). It is a free
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
R-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
generated by the elements 1 - g, for g in G.
Examples
Given a ring R and the (additive) monoid of the natural numberNatural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s N (or {xn} viewed multiplicatively), we obtain the ring R[{xn}] =: R[x] of polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s over R.
The Monoid Nn (with the addition) gives the polynomial ring with n variables: R[Nn] =: R[X1, ..., Xn].