Skip to content

2001 Nov

2001 November 26

  1. Let \(R\) be a ring and \(M\) a (left) \(R\)-module. Suppose that \(M = K\oplus L\). In this problem, maps are written on the right, opposite the scalars that operate on the left.\ a. Let \(\phi \in \operatorname{Hom}_R(L, K)\). Show that \(M = K \oplus L(1 + \phi)\) where \(L(1+\phi) = \{x + x\phi : x \in L\}\)\ b. Suppose that \(M = K \oplus L'\). show that there is \(\phi \in \operatorname{Hom}_R(L, K)\) such that \(L' = L(1 +\phi)\).\ c. Prove that \(L(1 + \phi) = L(1 + \psi)\) for \(\phi, \psi \in \operatorname{Hom}_R(L, K)\) if and only if \(\phi = \psi\).\ d. Note that the previous parts establish a bijective correspondence between complementary summands of \(K\) in \(M = K \oplus L\) and \(\operatorname{Hom}_R(L, K)\). Let \(M\) be a \(\mathbb Z\)-module (i.e., an abelian group), and suppose \(M = \langle a \rangle \oplus \langle b \rangle \oplus \langle c \rangle\), where \(\langle a \rangle\) is cyclic of prime order \(p\), \(\langle b \rangle\) is cyclic of order \(p^2\), and \(\langle c \rangle\) is cyclic of order \(p^3\). Count the different direct complements of \(\langle b \rangle \oplus \langle c \rangle\) in \(M\).

  2. Let \(p\) be a prime in \(\mathbb Z\), and let \(R = \{a/p^n : a, n \in \mathbb Z\}\).\ a. Prove that \(R\) is a PID, hence a unique factorization domain.\ b. What are the primes and what are the units in \(R\)?\ c. In case \(p=3\), prove that \(9\) is a greatest common divisor of \(5\) and \(7\).

  3. Let \(R\) be a ring and suppose that \(I, J\) are (two-sided) ideals of \(R\). Recall that \(IJ = \left\{\sum_i e_i f_i : e_i \in I, f_i \in J\right\}\).\ a. Prove that \(IJ\) is an ideal of \(R\) and \(IJ \subseteq I \cap J\).\ b. Assume now that \(R\) is a PID, \(I= Ra\), and \(J = Rb\). Show that \(IJ = RAB\) and that \(IJ = I\cap J\) if and only if \(ab\) is a least common multiple of \(a\) and \(b\).

  4. Explain why \(\left(\frac{\mathbb Z}{\mathbb 5Z}\right) [x] / (x^2 - 2)\) is a field.