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2004 Nov

2004 November

Notation: \(\mathbb Z\) denotes the ring of integers and \(\mathbb Q\) denotes the field of rational numbers.

  1. Let \(R\) be a commutative ring.\ a. Show that if \(R\) is an integral domain, then the only units in the polynomial ring \(R[x]\) are the units of \(R\).\ b. Give counterexamples when \(R\) is not an integral domain.\ c. Show that \(R[x]\) is a principal ideal domain if and only if \(R\) is a field.

  2. Let \(f\in \mathbb Z[x]\) be a monic polynomial of degree \(n\) with distinct roots \(\alpha_1, \dots, \alpha_r\), \(r \leq n\). Show that \(\alpha_1 + \cdots + \alpha_r \in \mathbb Z\).

  3. List all the ideals of the quotient ring \(\mathbb R[x]/I\), where \(I\) is the ideal generated by \((x-5)^2(x^2 + 1)\). Identify which of the ideals are prime and which are maximal. Does your answer change if the field \(\mathbb R\) is replaced by \(\mathbb C\)? (Explain.)

  4. Let \(M\) be a left module over a ring \(R\).\ a. Suppose that \(M\) is finitely generated and that \(R\) is commutative and Noetherian. Sketch a proof that \(M\) is Noetherian.\ b. Suppose that \(M\) is Noetherian and that \(f: M \rightarrow M\) is a surjective homomorphism. Show that \(f\) is an isomorphism.

  5. Let \(R\) be a ring with \(1\) and let \(M\) be an \(R\)-module. Show that the following are equivalent:\ i. There exists a module \(N\) such that \(M\oplus N\) is free.\ ii. Given any surjection \(\varphi: B\rightarrow M\), there exists an \(R\)-module homomorphism \(\psi: M \rightarrow B\) such that \(\varphi \circ \psi\) is the identity on \(M\).\ iii. Given a homomorphism \(\varphi: M\rightarrow B\) and a surjection \(\pi: A \rightarrow B\), there exists a homomorphism \(\psi : M \rightarrow A\) such that \(\pi \circ \psi = \varphi\).

  6. Let \(R = \mathbb Z[i]\) be the ring of Gaussian integers.\ a. Show that any nontrivial ideal must contain some positive integer.\ b. Find all the units in \(R\).\ c. If \(a + bi\) is not a unit, show that \(a^2 + b^2 > 1\).