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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]]

Revision as of 11:58, 1 March 2014

Apart from adding inches to your manhood, these pills also increase sexual stamina, orgasm and erection strength. If you are a man then you have thought about taking male enhancement supplements, especially if you are in a relationship. But as things and people progressed it was edged out beyond its uses and the result was spam. Penile bands help to hold the penis in position and prevent quick flow back of blood from the penis to the body. Penis enlargement had probably not been very well-known until a few decades ago.

This is the herb that has helped the 60, 70, 80 even 90 year olds in Brazil have a healthy sex life naturally. These exercises will grow both the length and girth of your penis and if you follow the guides and agenda, you will view noticeable results very quickly. It is advised to stop usage once this occurs to promote tissue health and safety. You see it is like this, the sensitive areas on a woman's vagina are at the opening. So far that best way to avoid falling prey to a scam site is to do your research.

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So if you are going for male enhancement products, look for male enhancement reviews at the earliest opportunity you get online. The first exercise to enlarge your penis an effective exercise would be to massage the penis and you can use a vibration type exercise that started in East Germany with a technique called rhythmic neuromuscular stimulation. And what, you curious pink-nosed kitten, is the median size of male genitalia. Search Sub Engines also move your listing to enable everyone to have the chance to reach that all important top result and pull in those profits which appears to be far more than we have been offered in the past. But what the majority of people do not get is that the penis is not just all muscle.

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