# Structure of module over Eilenberg MacLane spectrum

Let $$HR$$ be the Eilenberg-Maclane spectrum for a commutative ring $$R$$ and $$M$$ be a module over $$HR.$$ Then I want to prove that $$M$$ is a product of Eilenberg-Mac Lane spectra.

Construction: Let $$\pi_k(M)$$ be generated by a set $$F_k$$ for each $$k \geq 0.$$ Then we have a map

$$\vee_{k \geq 0} \vee_{a \in F_k} HR_a \to M$$.

Out of this, I need to find a structure of $$M.$$

Any suggestion will be appreciated.

• You still say you want to prove that a generic module $M$ is a product of copies of $HR$. But then in the formula you wrote coproducts. Are you trying to show that every module for any E-ML spectrum is free?
• In spectra, finite products and coproducts coincide. But module spectra over Eilenberg-Mac Lane spectra need not be free: e.g., $H\mathbb{Z}/2$ is an $H\mathbb{Z}$-module, but isn't a product of $H\mathbb{Z}$'s. This is only true if $R$ is a field.