Infinite direct sum of projective modules

Question

Let $A$ be a noncommutative algebra over a field $k$, and $M$ a left $A$-module such that $$ M = \bigoplus_{i \in I} M_i $$ where $I$ is a countable set, and the decomposition is a decomposition of left $A$-modules. If each $M_i$ is projective as a left $A$-module, can we conclude that $M$ is projective as a left $A$-module?

Answer 1

Yes, that is true.

Hint for showing that: Use the universal property of the direct sum in form of the bijection between the respective Hom-sets (or just in terms of the diagram if you prefer that).