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2003 Apr

2003 April

  1. Let \(R\) be a ring, \(I\) a right ideal and \(J\) a left ideal of \(R\). The expression \(I+J\) stands for the Abelian group generated by \(I\cup J\). Prove that

    \[\frac{R}{I}\otimes_R \frac{R}{J} \cong \frac{R}{I+J}.\]

    Define your maps carefully with particular attention to well-definedness.

  2. Let \(A_1\) and \(A_2\) be modules over some ring, \(K_1\subseteq A_1\) and \(K_2\subseteq A_2\) be submodules, and \(\gamma_1: A_1\rightarrow A_2\) and \(\gamma_2: A_2\rightarrow A_1\) and be homomorphisms with the following properties.\ i. \(\gamma_1(K_1) \subseteq K_2\) and \(\gamma_2(K_2) \subseteq K_1\),\ ii. \((1-\gamma_2\gamma_1)(A_1) \subseteq K_1\) and \((1-\gamma_1\gamma_2)(A_2) \subseteq K_2\).\ a. Show that the maps \(f \colon A_1\oplus K_2 \rightarrow A_2\oplus K_1\) and \(g \colon A_2\oplus K_1 \rightarrow A_1\oplus K_2\) defined by \begin{align} f(x_1,y_2) &= (\gamma_1(x_1) - y_2, -x_1 + \gamma_2\gamma_1(x_1) - \gamma_2(y_2)), \text{ and }\\ g(x_2,y_1) &= (\gamma_2(x_2) - y_1, -x_2 + \gamma_1\gamma_2(x_2) - \gamma_1(y_1)) \end{align} are well-defined homomorphisms and that \(g = f^{-1}\).\ b. Let \(P_1\) and \(P_2\) be projective modules and \(M\) a module over some ring \(R\). Suppose that \(\beta_1: P_1\rightarrow M\) and \(\beta_2: P_2 \rightarrow M\) are epimorphisms. Show that \(P_1\oplus \ker(\beta_2) \cong P_2\oplus \ker(\beta_1)\).

  3. Let \(M_n(R)\) be the ring of matrices with coefficients in a ring \(R\) (with 1), where \(n\) is a natural number \(> 1\). For a (two-sided) ideal \(I\) of \(R\), let \(M_n(I) = \{[a_{ij}]\in M_n(R) : a_{ij}\in I\}\).\ a. Prove that \(M_n(I)\) is an ideal of \(M_n(R)\).\ b. Prove that every ideal of \(M_n(R)\) is of the form \(M_n(I)\) for a suitable ideal \(I\) of \(R\).

  4. Recall that a module is simple if it is nontrivial and has no proper submodules.\ a. Prove that the endomorphism ring of a simple module is a division ring (= skew field).\ b. Prove that a ring \(R\) with identity 1 contains a left ideal \(I\) such that \(R/I\) is a simple left \(R\)-module.\ c. Let \(R\) be a ring with identity 1 and suppose that \(M = _RM\) is a left \(R\)-module with the property that every submodule is a direct summand (“everybody splits”). Show that every non-zero submodule \(K\) of \(M\) contains a simple submodule.