definition of product of modules

Question

I have been given this definition of a product between modules:

If $I$ is an indexing set with $M_i$ as an $R$-Module then the product $\prod \limits_{i \in I} M_i$ is defined as the set consisting of I indexed tuples $(x_i)_{i\in I}$ for each $i \in I$ which is made into an R module by component wise addition and multiplication.

my problem is understanding this definition, could anyone give me a basic example of what the product of two Modules $M_1$ and $M_2$ would be, just so i could see how it works practically.

thanks in advance for the help!

Answer 1

Consider $\prod \limits_{i \in I} M_i$.

The sum is $$( a_i)_{i \in I } + ( b_i)_{i \in I } = ( a_i + b_i)_{i \in I }$$ The action of R : $$ r \cdot ( a_i)_{i \in I } = ( r \cdot a_i)_{i \in I }$$