abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

, a monoid ring is a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

constructed from a ring and a

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...

, just as a

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 the set of elements of the give ...

is constructed from a ring and a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

Definition

Let ''R'' be a ring and let ''G'' be a monoid. The monoid ring or monoid algebra of ''G'' over ''R'', denoted ''R'' 'G''or ''RG'', is the set of formal sums

\sum_ r_g g

, where

r_g \in R

for each

g \in G

and ''r''_''g'' = 0 for all but finitely many ''g'', equipped with coefficient-wise addition, and the multiplication in which the elements of ''R'' commute with the elements of ''G''. More formally, ''R'' 'G''is the set of functions such that is finite, equipped with addition of functions, and with multiplication defined by :

(\phi \psi)(g) = \sum_ \phi(k) \psi(\ell)

. If ''G'' is a

, then ''R'' 'G''is also called the

of ''G'' over ''R''.

Universal property

Given ''R'' and ''G'', there is a

ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...

sending each ''r'' to ''r''1 (where 1 is the identity element of ''G''), and a

monoid homomorphism In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...

(where the latter is viewed as a monoid under multiplication) sending each ''g'' to 1''g'' (where 1 is the multiplicative identity of ''R''). We have that α(''r'') commutes with β(''g'') for all ''r'' in ''R'' and ''g'' in ''G''. The universal property of the monoid ring states that given a ring ''S'', a ring homomorphism , and a monoid homomorphism to the multiplicative monoid of ''S'', such that α'(''r'') commutes with β'(''g'') for all ''r'' in ''R'' and ''g'' in ''G'', there is a unique ring homomorphism such that composing α and β with γ produces α' and β '.

Augmentation

The augmentation is the ring homomorphism defined by :

\eta\left(\sum_ r_g g\right) = \sum_ r_g.

The

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...

of ''η'' is called the augmentation ideal. It is a

free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...

''R''-

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

with basis consisting of 1 – ''g'' for all ''g'' in ''G'' not equal to 1.

Examples

Given a ring ''R'' and the (additive) monoid of

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

s N (or viewed multiplicatively), we obtain the ring ''R''[] =: ''R''[''x''] of polynomials over ''R''. The monoid N^''n'' (with the addition) gives the polynomial ring with ''n'' variables: ''R''[N^''n''] =: ''R''[''X''₁, ..., ''X''_''n''].

Generalization

If ''G'' is a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

, the same construction yields a semigroup ring ''R'' 'G''

References

*{{cite book , first = Serge , last = Lang , authorlink=Serge Lang , title = Algebra , publisher =

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, location = New York , year = 2002 , edition = Rev. 3rd , series =

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard s ...

, volume=211 , isbn=0-387-95385-X