{{short description|Algebraic structure}} In abstract algebra, a '''monoid ring''' is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group.

==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 <math>\sum_{g \in G} r_g g</math>, where <math>r_g \in R</math> for each <math>g \in G</math> and ''r''<sub>''g''</sub> = 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 free ''R''-module on the set ''G'', endowed with ''R''-linear multiplication defined on the base elements by ''g·h'' := ''gh'', where the left-hand side is understood as the multiplication in ''R''[''G''] and the right-hand side is understood in ''G''.

Alternatively, one can identify the element <math> g \in R[G] </math> with the function ''e<sub>g</sub>'' that maps ''g'' to 1 and every other element of ''G'' to 0. This way, ''R''[''G''] is identified with the set of functions {{nowrap|φ: ''G'' → ''R''}} such that {{nowrap|{''g'' : φ(''g'') ≠ 0}}} is finite. equipped with addition of functions, and with multiplication defined by :<math> (\phi \psi)(g) = \sum_{k\ell=g} \phi(k) \psi(\ell)</math>. If ''G'' is a group, then ''R''[''G''] is also called the group ring of ''G'' over ''R''.

==Universal property== Given ''R'' and ''G'', there is a ring homomorphism {{nowrap|α: ''R'' → ''R''[''G'']}} sending each ''r'' to ''r''1 (where 1 is the identity element of ''G''), and a monoid homomorphism {{nowrap|β: ''G'' → ''R''[''G'']}} (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 {{nowrap|α': ''R'' → ''S''}}, and a monoid homomorphism {{nowrap|β': ''G'' → ''S''}} 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 {{nowrap|γ: ''R''[''G''] → ''S''}} such that composing α and β with γ produces α' and β '.

==Augmentation==

The augmentation is the ring homomorphism {{nowrap|''&eta;'': ''R''[''G''] → ''R''}} defined by : <math> \eta\left(\sum_{g\in G} r_g g\right) = \sum_{g\in G} r_g.</math>

The kernel of ''&eta;'' is called the augmentation ideal. It is a free ''R''-module with basis consisting of 1&nbsp;–&nbsp;''g'' for all ''g'' in ''G'' not equal to 1.

== Examples ==

Given a ring ''R'' and the (additive) monoid of natural numbers '''N''' (or {''x''<sup>''n''</sup>} viewed multiplicatively), we obtain the ring ''R''[{''x''<sup>''n''</sup>}] =: ''R''[''x''] of polynomials over ''R''. The monoid '''N'''<sup>''n''</sup> (with the addition) gives the polynomial ring with ''n'' variables: ''R''['''N'''<sup>''n''</sup>] =: ''R''[''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>].

==Generalization== If ''G'' is a semigroup, the same construction yields a '''semigroup ring''' ''R''[''G''].

==See also== * Free algebra * Puiseux series

== References == *{{cite book | first = Serge | last = Lang | authorlink=Serge Lang | title = Algebra | publisher = Springer-Verlag | location = New York | year = 2002 | edition = Rev. 3rd | series = Graduate Texts in Mathematics | volume=211 | isbn=0-387-95385-X}}

== Further reading == *R.Gilmer. ''[https://books.google.com/books?id=sOPNfkp-Le8C&q=%22monoid+ring%22 Commutative semigroup rings]''. University of Chicago Press, Chicago–London, 1984

Category:Ring theory