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In [[abstract algebra]], a '''monoid ring''' is a [[ring (algebra)|ring]] constructed from a ring and a [[monoid]], just as a [[group ring]] is constructed from a ring and a [[group (mathematics)|group]].
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==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 G</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 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 (mathematics)|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(\sum_{g\in G} r_g g) = \sum_{g\in G} r_g.</math>
 
The [[kernel (set theory)|kernel]] of ''&eta;'' is called the '''augmentation ideal'''. It is a [[free module|free]] ''R''-[[module (mathematics)|module]] 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]]s '''N''' (or {''x''<sup>''n''</sup>} viewed multiplicatively), we obtain the ring ''R''[{''x''<sup>''n''</sup>}] =: ''R''[''x''] of [[polynomial]]s 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 rng''' ''R''[''G''].
 
==See also==
* [[Free algebra]]
 
== 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. ''Commutative semigroup rings''. University of Chicago Press, Chicago–London, 1984
 
[[Category:Ring theory]]

Latest revision as of 06:25, 5 January 2015

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