Integral Closure

Prove that \(\mathbb {C}[x]\) is integrally closed in \(\mathbb {C}(x)\), the field of rational functions.

1. List the prime ideals of the ring \(R=\mathbb {Z}[x,y]/\langle 3+x, y(y-1)+x^2, x\rangle \).
2. Give an example of an integral domain with exactly 2 prime ideals.

Let \(\mathbb {Z}[x]\) be the ring of polynomials in \(x\) with integer coefficients. Let \(R\) denote the subring consisting of all polynomials having their coefficients of \(x\) and \(x^2\) equal to 0.

1. Show that \(\mathbb {Q}(x)\) is the field of fractions of \(R\).
2. Compute the integral closure of \(R\) in \(\mathbb {Q}(x)\).
3. Does there exist a polynomial \(g(x)\in R\) such that \(R\) is generated as a ring by 1 and \(g(x)\)?

Let \(R\) be a commutative integral domain with identity. Recall that an \(R\)-module is torsion free if for any \(r\in R\) and \(m\in M\), \(rm=0\) implies that either \(r=0\) or \(m=0\).

1. Suppose \(R\) is a principal ideal domain and \(M\) and \(N\) are two \(R\) modules. Prove that if \(M\) and \(N\) are torsion free, so is \(M\otimes _R N\).
2. Give an example of an integral domain \(R\) and two torsion free \(R\)-modules \(M\) and \(N\) such that \(M\otimes _R N\) is not torsion free.

Let \(R\) be a commutative ring with identity. Let \(G\) be a finite subgroup of \(R^\times \), the group of units of \(R\). Prove that if \(R\) is an integral domain, then \(G\) is cyclic.