Simple examples for the coproduct and product of the family of R-modules $\left\{M_{\alpha }\right\}_{\alpha \in A}$ I know the definition (for the coproduct it's a module M with the family of R-homomorphisms $w_{\alpha }:M_{\alpha }\rightarrow M$ such that for every R-module N and the family of R-homomorphisms $f_{\alpha }:M_{\alpha }\rightarrow N$ there exists exactly one $f:M\rightarrow N$ such that $f\left(w_{\alpha }\right)=f_{\alpha }$ for all alpha
The definition for the product is basically the opposite of that (from M to $M_{alpha}$, from N to $M_{alpha}$, and from N to M)
Could anyone show me some simple examples for the coproduct and product?
 A: Well, there is a standard, down-to-earth construction of products and coproducts of modules using the language of set theory: given an indexed family $\{M_\alpha\}_{\alpha\in A}$ of modules, the product is the set-theoretic product $$\prod_{\alpha\in A}M_\alpha=\left\{x: A\to \cup_{\alpha\in A}M_\alpha\,\middle\vert\,\forall \alpha\in A, x(\alpha)\in M_\alpha\right\}$$
endowed with componentwise addition and product by scalar, i.e. $x+y$ is the function $x+y:A\to\cup_{\alpha\in A}M_\alpha$ such that $(x+y)(\alpha)=x(\alpha)+y(\alpha)$ for all $\alpha$, and for $\lambda\in R$ the element $\lambda x$ is the function such that $(\lambda x)(\alpha)=\lambda x(\alpha)$ for all $\alpha$. The projections $v_\beta:\prod_{\alpha\in A}M_\alpha\to M_\beta$ are the evaluations $f_\beta(x)=x(\beta)$.
The coproduct may be taken to be the submodule $$\bigoplus_{\alpha\in A}M_\alpha=\left\{x\in\prod_{\alpha\in A}M_\alpha\,\middle\vert\, \operatorname{card}\{\beta\in A\mid x(\beta)\ne 0\}<\aleph_0\right\}$$
with the injections being the maps $w_\beta:M_\beta\to \bigoplus_{\alpha\in A}M_\alpha$, $$(w_\beta(v))(\alpha)=\begin{cases}v&\text{if }\alpha=\beta\\ 0&\text{if }\alpha\ne\beta\end{cases}$$
In other words, $\bigoplus_{\alpha\in A}M_\alpha$ is the set of the elements of the product that evaluate to zero outside of some finite subset of $A$ (which needs not be the same for all the elements).
As an exercise, you can test your understanding with $R=\Bbb Z$, $A=\Bbb N$ and $M_\alpha=(\Bbb Z/\alpha\Bbb Z)^\alpha$.
