Exercise 2, § 1

Let E be a left A-module; for every left ideal $a$ of A let $aE$ denote the sum of the $ax$ where $x \in E$, which is therefore a submodule of E.

(a) Show that for every family $(E_\lambda)_{\lambda \in L}$ of A-modules, $a \cdot \bigoplus_{\lambda \in L} E_\lambda$ is canonically identified with $\bigoplus_{\lambda \in L} aE_\lambda$.

(b) If $a$ is a finitely generated left ideal of A, show that for every family $(E_\lambda)_{\lambda \in L}$ of A-modules, $a \cdot \prod_{\lambda \in L} E_\lambda$ is canonically identified with $\prod_{\lambda \in L} aE_\lambda$. Give an example where $a$ is not finitely generated and the above property fails to hold (take A commutative and all the $E_\lambda$ equal to A).

(c) Let K be a commutative field, A the ring $K[X, Y]$ of polynomials in two indeterminates over K, $m = AX$ and $n = AY$. If E is the quotient A-module of $A^2$ by the submodule of $A^2$ generated by $Xe_1 - Ye_2$ (where $(e_1, e_2)$ is the canonical basis of $A^2$), show that $(m \cap n)E \neq (mE) \cap (nE)$