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2008 Apr

2008 April

  1. Let \(R\) be a commutative ring.\ a. Define what it means for \(R\) to be Noetherian.\ b. Prove that if \(R\) is Noetherian, the polynomial ring \(R[x]\) is Noetherian.

  2. Let \(R\) be a commutative ring with identity. Recall that an ideal \(P\subset R\) is prime if \(P\neq R\) and \(ab \in P \Rightarrow a\in P\) or \(b\in P\). Let \(J\subseteq R\) be an ideal.\ a. Show that \(J\) is prime if and only if \(R/J\) is an integral domain.\ b. Show that \(J\) is maximal if and only if \(R/J\) is a field.\ c. Prove that if \(J\neq R\), then \(J\) is contained in a maximal ideal.\ d. Prove that an element of \(R\) is nilpotent if and only if it belongs to every prime ideal of \(R\).

  3. Let \(R\) be a ring with identity.\ a. Suppose that \(A\) is a right \(R\)-module and \(B\) is a left \(R\)-module. Define the abelian group \(A\otimes_R B\).\ b. Show that \(\mathbb Z_m \otimes \mathbb Z_n \cong \mathbb Z_d\), where \(d = \gcd(m,n)\).