Show that $I=R[x]/\langle P \rangle$ is a free R-Module and finite generated R-Algebra for $P\in R[x]$ a polynomial of degree d, R an integral domain and $P$ normed(scaled?). What would happen if it wouldn't be normed? Would it also be an integral domain?
By normed I mean that for $P=\sum_{i=0}^{d}p_{i}x^{i}$ then $p_{d}=1$.This would mean that the leading coefficient of P is 1. This would mean that the leading coefficient is in the integral domain(?) R*=(R[x])*. And well we have learned a theorem that if this is the case then it is a free R-Module.
Is this it? Do I have now to show only that it is also a R-Algebra? Where exactly am I using the part that it is normed? How do I know that the leading coefficient is indeed in the integral domain and also what about I being an integral domain itself?
