Due: Friday, Oct 10, 2008
Math 371 / Problem Set 4 (two pages)
Polynomial functions
Let R⊂Sbe rings with identity, and R[X] be the ring of polynomials in the variable Xover R. Further
let F(S, S) be the ring of all the maps from f:S→Sas introduced at Problem 6 of Problem Set 3.
1) Prove / answer the following:
a) For every α∈Sthere exists a unique ring homomorphism φα:R[X]→Ssuch that φα(a) = afor all
a∈R, and φα(X) = α.
b) What is Ker(φα) for α= 0 and α= 1? And what is Ker(φα) for arbitrary α∈R?
[Hint: Prove by induction that φα(Xm) = αmfor all m; hence φα(aXm) = aαmfor all a∈Rand all m;
hence by induction, if p(X) = a0+a1X+. . . +anXn, then φαp(X)=a0+a1α+. . . +anαn, etc.]
Language: If p(X)∈R[X] is a polynomial, we denote p(α) := a0+a1α+. . . +anαn=φαp(X), and call
it the value or the evaluation of p(X) at X=α.
Define φ:R[X]→ F(S, S ), by p(X)7→ fp(X), where fp(X):S→S,α7→ fp(X)(α) := p(α) for all α∈S.
Language: The function fp(X):S→Sdefined by fp(X)(α) := p(α) for all α∈S, is called the polynomial
function on Sdefined by p(X).
2) In the above notations, answer the following:
a) Show that φis a ring homomorphism.
b) Analyze the injectivity/surjectivity of φ:R[X]→ F(R, R ) in the following cases:
i) R=Q,R=R,R=C.
ii) R=Z/m Zfor m= 2,3,4,5,6,7.
Modules and vector spaces
3) Let Mbe an R-module, and N⊆Ma non-empty subset. Prove that the following assertions are
equivalent:
i) Nis an R-submodule of M.
ii) For all x, y ∈Nand r∈Rone has: x−y∈Nand rx ∈N.
iii) For all x, y ∈Nand r, s ∈Rone has: rx +sy ∈N.
4) Let M1, M2be R-modules, and consider M:= M1×M2viewed as an abelian group w.r.t. the component
wise addition. Prove/answer the following:
a) Via the outer multiplication r(x1, x2) := (rx1, rx2), Mbecomes an R-module.
b) Let Ni⊂Mi,i= 1,2, be subsets, and set N:= N1×N2viewed as subset of M=N1×N2. Then
N⊂Mis an R-submodule iff N1⊆M1and N2⊆M2are so.
c) Is the same correspondingly true, for nmodules M1, . . . , Mnand their product M=M1×. . . ×Mn
endowed with the component wise addition?
5) Answer the following:
a) Let Vbe an F-vector space. Show that for all α∈Fand v∈Vone has: αv = 0 iff α= 0 or v= 0.
b) Is the same true for all rings Rwith 1R6= 0Rand all R-modules M?
6) Let Pol(R)⊂ F(R,R) be the set of all the polynomial functions. Prove or disprove:
a) The set V0={f∈ Pol(R)|f(0) = 0}is an R-vector space, which is not finitely generated.
b) Is the same is true for the set V1={f∈ Pol(R)|f(1) = 1}?
