= Solution
If $R$ is left Noetherian and $I\to\bigoplus_\alpha E_\alpha$ is a map from a left ideal, finitely many generators of $I$ have support in one finite set of summands. The map therefore lands in a finite direct sum of injectives and extends to $R$. Baer's criterion proves that the full direct sum is injective.
Conversely, let $I_1\subseteq I_2\subseteq\cdots$ and put $I=\bigcup I_n$. Embed each $R/I_n$ in an injective module $E_n$. The map
$$
I\longrightarrow\bigoplus_{n\geq1}E_n,\qquad
a\longmapsto(a+I_n)_n
$$
has finite support. If the direct sum is injective, it extends to $R$; the extension's value at $1$ has finite support, forcing $a\in I_n$ for every sufficiently large $n$ and every $a\in I$. Hence the chain stabilizes. This is the <Bass-Papp theorem>.
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