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?


Since $P_1$ and $P_2$ are given only as $R$-modules, and not as submodules of some pre-existing larger $R$-module containing both of them, it only makes sense initially for $P_1 \oplus P_2$ to be an external direct sum. But then $P_1$ and $P_2$ have canonical isomorphic copies inside $P_1 \oplus P_2$, namely $P_1 \times \{0\}$ and $\{0\} \times P_2$, and $P_1 \oplus P_2$ is an internal direct sum of these isomorphic copies.


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