Monoid ring
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[edit]
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 , where for each and rg = 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 with the function eg that maps g to 1 and every other element of G to 0. This way, R[G] is identified with the set of functions
- .
If G is a group, then R[G] is also called the group ring of G over R.
Universal property[edit]
Given R and G, there is a ring homomorphism
The universal property of the monoid ring states that given a ring S, a ring homomorphism
Augmentation[edit]
The augmentation is the ring homomorphism
The kernel of
Examples[edit]
Given a ring R and the (additive) monoid of natural numbers N (or {xn} viewed multiplicatively), we obtain the ring R[{xn}] =: R[x] of polynomials over R. The monoid Nn (with the addition) gives the polynomial ring with n variables: R[Nn] =: R[X1, ..., Xn].
Generalization[edit]
If G is a semigroup, the same construction yields a semigroup ring R[G].
See also[edit]
References[edit]
- Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (Rev. 3rd ed.). New York: Springer-Verlag. ISBN 0-387-95385-X.
Further reading[edit]
- R.Gilmer. Commutative semigroup rings. University of Chicago Press, Chicago–London, 1984