= Solution
The universal property of the <polynomial ring> gives the unique complex <algebra homomorphism>
$$
\iota:\mathbb C[Y]\longrightarrow\mathbb C[X],\qquad
\sum_jc_jY^j\longmapsto\sum_jc_jX^{2j}.
$$
Distinct monomials $X^{2j}$ are linearly independent, so $\iota$ is injective. <Restriction of scalars> defines $b\cdot m=\iota(b)m$ for $b\in B$ and $m\in M$; its <module> axioms follow from those of the original action and the <algebra homomorphism>. In particular \b[$Y$ acts as the square of the operator defining the $X$ action].
Back to article page
