Completely lost when reading this:
"Defintion: If the index set $T$ is finite, the coproduct and product are the same module. If $T$ is infinite, the coproduct or sum $$\coprod_{t\in T}M_t=\bigoplus_{t\in T}M_t=\oplus M_t$$ is the submodule of $\prod M_t$ consisting of all sequences $\{m_t\}$ with only a finite number of non-zero terms ..."
I think I understand what the product of module is. It's just like $n$-tuple $\{m_1,m_2,\cdots,m_n\}$ when $T={1,2,\cdots,n}$, but generalized to general index set $T$, right?
But what are coproducts? Submodule with finite number of non-zero terms only? Why need to restrict it to finite number? Why it is also called a sum?