Exercise 26, § 8

(a) In the notation of Proposition 19 (no. 10), suppose that there exist two $A[X]$-linear mappings $g_1, g_2$ of $M[X]$ into $M'[X]$ such that $\psi' \circ g_2 = g_1 \circ \psi$; show that there then exists an $A[X]$-linear mapping $g$ of $M_u$ into $M'_u$, such that $\phi' \circ g_1 = \phi' \circ \bar{g}$. Further, if $g_1$ and $g_2$ are $A[X]$-isomorphisms of $M[X]$ onto $M'[X]$, $g$ is an $A[X]$-isomorphism of $M_u$ onto $M'_u$.

(b) In the notation of no. 10, the endomorphisms $u, u'$ are called equivalent if there exist two isomorphisms $f_1, f_2$ of $M$ onto $M$ such that $u' \circ f_2 = f_1 \circ u$; they are called similar if there exists an isomorphism $f$ of $M$ onto $M'$ such that $u' \circ f = f \circ u$. Deduce from (a) that, for $u$ and $u'$ to be similar, it is necessary and sufficient that the endomorphisms $X - \bar{u}$ and $X - \bar{u}'$ (of the $A[X]$-modules $M[X]$ and $M'[X]$ respectively) be equivalent.