Let $M, M_1,\ldots,M_k$ be $R$-modules and let $N_1,\ldots,N_k$ be submodules of the $R$-module $M$.
The authors Dummit and Foote state:
"The direct product of $M_1,\ldots,M_k$ is also referred to as the (external) direct sum of $M_1,\ldots,M_k$, and is denoted $M_1\oplus \cdots\oplus M_k$."
Then they state a theorem, which I understand, and some more stuff and then they state:
"If $M=N_1 + \cdots +N_k$ satisfies the equivalent conditions of the proposition above then $M$ is said to be the (internal) direct sum of $N_1,\ldots,N_k$ and is denoted by $M=N_1 \oplus \cdots \oplus N_k$."
In the exercises, problem 3 says: "Let $P_1$ and $P_2$ be $R$-modules. Prove that $P_1 \oplus P_2$ is projective if and only if $P_1$ and $P_2$ are both projective."
Question: Is $P_1 \oplus P_2$ an internal or external direct sum?