习题 16, § 5

令$E$和$F$为交换环$k$之二模，并令$p$为整数$\geq 0$。

a) 证明如果$\mathrm{Hom}(\Gamma_p(E), F)$的$p \leq 2$、映射$\lambda_p$进入$F^E$，与$\sigma$映射$\sigma \circ \gamma_p$相关联，是单射映射。 b) 证明$A$有像$\mathrm{Hom}_k(E, F)$。 Show that $A$, has as image the set of all mappings $g$ of $E$ into $F$ such that : (i) $g(ax) = a^2g(x)$ for any $a \in k$ and $x \in E$; (ii) the mapping $(x, y) \mapsto g(x + y) - g(x) - g(y)$ is a bilinear mapping of $E \times E$ into $F$. c) Show that if $E$ is a free $k$-module, then the image of $\lambda_p$ is the module $\mathrm{Pol}_k^p(E, F)$ of homogeneous polynomial mappings of degree $p$ of $E$ into $F$.