Monoid Ring - Definition

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) = Σkl=gφ(k)ψ(l). The set of all such functions φ, together with these two operations, forms a ring, the monoid ring of G over R denoted R. If G is a group, then R denotes the group ring of G over R.

Less rigorously but more simply, an element of R 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 R-product and gihj is the G-product.

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

T(r)(1G) = r, T(r)(g) = 0 for g ≠ 1G.

where 1G is the identity element of G.

There also exists a canonical homomorphism going the other way, called the augmentation. It is the map ηR:R → R, defined by

The kernel of this homomorphism, the augmentation ideal, is denoted by JR(G). It is a free R-module generated by the elements 1 - g, for g in G.

Read more about this topic:  Monoid Ring

Famous quotes containing the word definition:

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)