Let $\left\{ f_{i}:M_{i}\longrightarrow N\right\} _{i\in I}$, can $Im\left(\oplus_{i\in I}f_{i}\right)=\sum_{i\in I}Im\left(f_{i}\right)$

Question

Let $\left\{ f_{i}:M_{i}\longrightarrow N\right\} _{i\in I}$ be a family of $R$-homorphism from $R$-module $M_{i}$ to $R$-module $N$. Do we have $Im\left(\oplus_{i\in I}f_{i}\right)=\sum_{i\in I}Im\left(f_{i}\right)$?

Answer 1

Yes, $x \in \sum_iim(f_i)$ if and only if you can write $x = \sum_if_i(m_i)$ where $m_i \in M_i$ and only finitely many $m_i$ are nonzero. But this is equivalent to $m = (m_i)_i \in \bigoplus_iM_i$ and $x = \left(\bigoplus_if_i\right)(m)$.