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Monoid RingIn abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid. Let R be a ring and G be a monoid. We can look at all the functions φ : G -> R such that the set {g: φ(g) ≠ 0} is finite. We can define addition of such functions to be element-wise additions. We can define multiplication by (φ * ψ)(g) = Σkl=gφ(k)ψ(l). The set of all these functions, together with these two operations, forms a ring, the monoid ring of R over G; it is denoted by RG. If G is a group, then it is called the group ring of R over G. Put less rigorously but more simply, an element of RG is a polynomial in G over R, hence the notation. We multiply elements as polynomials, taking the product in G of the "indeterminates" and gathering terms: -
where risj is the product in R and gihj is the product in G. The ring R can be embedded into the ring RG via the ring homomorphism T: R->RG defined by - T(r)(1G) = r, T(r)(g) = 0 for g ≠ 1G.
where 1G denotes the identity element in G. There is also a canonical homomorphism going the other way; the augmentation is the map ηR:RG -> R defined by -
The kernel of this homomorphism is called the augmentation ideal and is denoted by JR(G). It is a free R-module generated by the elements 1 - g, for g in G. Examples Given a ring R and the monoid of the non-negative integers, N ({xn} viewed multiplicatively), we obtain the ring R{xn} =: Rx of polynomials over that ring.
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