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1995 Apr

1995 April

  1. Let \(R\) be a commutative ring. Recall that the annihilator of an element \(x\) in an \(R\)-module is the ideal \(\mathrm{ann}(x) = \{s \in R : sx = 0\}\). Let \(Rx\) and \(Ry\) be cyclic \(R\)-modules.\ a. Show that \(Rx \otimes_R Ry = R(x \otimes y)\).\ b. Show that \(R(x \otimes y) \cong R/(\mathrm{ann}(x) + \mathrm{ann}(y))\).\ c. Show that \(\mathrm{ann}(x \otimes y) = \mathrm{ann}(x) + \mathrm{ann}(y)\).\ (See also 1999 Nov, Prob 4.)

  2. Let \(R\) be a ring and suppose that \(I, J\) are left ideals of \(R\) such that \(I + J = R\). Prove that

    \[\frac{R}{I\cap J} \cong \frac{R}{I} \oplus \frac{R}{J}.\]

    (See also 1995 Nov, Prob 4.)

  3. Recall that an ideal \(I\) of a ring \(R\) is nilpotent if \(I^n = 0\) for some positive integer \(n\). Show: If \(I\) and \(J\) are nilpotent ideals, then so is \(I + J\). Let \(R = \mathbb Q[X,Y]/(X^2, Y^2)\). Show that \(R\) contains a unique largest nilpotent ideal and find it. (See also 1997 Nov, Prob 5.)

  4. The injectivity class of an \(R\)-module \(C\) is defined to be the class \(\operatorname{Inj}(C)\) consisting of all modules \(B\) such that for every submodule \(A\) of \(B\), every \(R\)-homomorphism \(h: A\rightarrow C\) extends to a homomorphism \(h' : B\rightarrow C\). Prove that the injectivity class of \(C\) is closed under epimorphic images; i.e., if \(B \in \operatorname{Inj}(C)\) and \(f : B \rightarrow B'\) is an epimorphism, then \(B'\in \operatorname{Inj}(C)\).

  5. Let \(M\) and \(N\) be finitely generated modules over the polynomial ring \(\mathbb Q[x]\). Show that \(M\) and \(N\) are isomorphic if \(M\) is isomorphic to a direct summand of \(N\) and \(N\) is isomorphic to a direct summand of \(M\). (See also 1997 Nov, Prob 2.)