show that the intersection of k[X^-1] and k[X] in k[X][X^-1] is only k

[MatthiasHu]

Two different approaches seem possible:

(1) "By comparing coefficients in R[X]": Consider the involution f : R[X][X^-1] --> R[X][X^-1]. We want to show that every polynomial p(X) which is invariant under f such that f(p(X)) = p(X^-1) is again a polynomial r(X) is constant. Let p(X) be of degree at most m. Then reversing the m+1 coefficients of p(X) yields a new polynomial q(X) of degree at most m such that X^m p(X^-1) = q(X) in the ring R[X][X^-1]. (This might be nasty?) Then X^m r(X) = q(X) in R[X], and we can compare coefficients to see that p is constant.

(2) "By comparing coefficients in R[X][X^-1]": Show that the underlying type of R[X][X^-1] is the type of merely-finite-support functions Z --> R. Show that the involution f : R[X][X^-1] --> R[X][X^-1] is induced by flipping Z. (Is this hard?) Then the claim follows by comparing coefficients of p(X) and p(X^-1).

[MatthiasHu]

About (2): We just fantasized that it might be best to construct general monoid rings and prove things about them first. (The monoid ring of a localization of a monoid M is a certain localization of the monoid ring of M).

To construct monoid rings, one might have to define a notion of merely-finite sum in a ring (or in an abelian group) first.

[MatthiasHu]

construct general monoid rings

Actually, monoid rings of discrete monoids... This is where the construction using "functions with merely finite support" works. In an arbitrary monoid ring you can not "lookup coefficients".

[felixwellen]

I just realized a notable alternative: Instead of comparing coefficients in R[X][X^-1], we can also compare coefficients in R[X][Y], which should be a lot easier, but then we have to conclude, that for P in R[X] and Q in R[Y], any S in R[X][Y] such that P-Q = S (XY-1) we get S=0. Then the result follows from: The intersection of R[X] and R[Y] in R[X][Y] is R.