Monoid ring - Citizendia

In abstract algebra, a monoid ring is a new ring constructed from some other ring and a monoid. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation

Definition

Let R be a ring and G be a monoid. Consider all the functions φ : G → R such that the set {g: φ(g) ≠ 0} is finite. Let all such functions be element-wise addable. We can define multiplication by (φ * ψ)(g) = Σ_kl=gφ(k)ψ(l). The set of all such functions φ, together with these two operations, forms a ring, the monoid ring of R over G denoted R[G]. If G is a group, then R[G] denotes the group ring of R over G. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively

Less rigorously but more simply, an element of R[G] is a polynomial in G over R, hence the notation. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations We multiply elements as polynomials, taking the product in G of the "indeterminates" and gathering terms:

$(\Sigma_i r_i g_i) \cdot (\Sigma_j s_j h_j) = \Sigma_{i,j} r_i s_j (g_i h_j),$

where r_is_j is the R-product and g_ih_j is the G-product. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

The ring R can be embedded in the ring R[G] via the ring homomorphism T : R → R[G] defined by

T(r)(1_G) = r, T(r)(g) = 0 for g ≠ 1_G. In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication

where 1_G is the identity element of G. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that

There also exists a canonical homomorphism going the other way, called the augmentation. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector It is the map η_R:R[G] → R ,defined by

$\sum_{g\in G} r_g g \rightarrow \sum_{g\in G} r_g$

The kernel of this homomorphism, the augmentation ideal, is denoted by J_R(G). In Mathematics, the kernel of a function f may be taken to be either the Equivalence relation on the function's domain It is a free R-module generated by the elements 1 - g, for g in G. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings

Examples

Given a ring R and the monoid of the natural numbers N ({xⁿ} viewed multiplicatively), we obtain the ring R[{xⁿ}] =: R[x] of polynomials over R. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

References

Lang, Serge (2002). Algebra, (Rev. 3rd ed. ), Graduate Texts in Mathematics 211, New York: Springer. ISBN 0-387-95385-X.

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