What means " Tor functor commutes with arbitary direct sum in both vanishing"?

Question

According to the following the video clip (at 39:11),

$\operatorname{Tor}$ functor commutes with arbitrary direct sum in both vanishing. ...... (★),

but I do not understand this property (I hope to see a specific example to satisfy (★) for the sake of my understanding).

More specifically,

$$\operatorname{Tor}_{R}^{i}(-,-) \oplus M =M \oplus \operatorname{Tor}_{R}^{i}(-,-)$$ (where $M$ is a ??? )

Then, I am not sure :

• which object $M$ is a proper one that commutes.(by gueswork, $M$ would be $R$-module)?
• what is meaning of direct sum in both vanishing?

Answer 1

It says in both variables. This means that for the situation where

1. $R$ is a ring
2. $M = \bigoplus_{j\in J} M_j$ is a right $R$-module that is the direct sum of submodules,

there is a natural map

$$\mathrm{Tor}^R_n(M,N) \longrightarrow \bigoplus_{j\in J}\mathrm{Tor}^R_n(M_j,N)$$

that is an isomorphism, and the same is true for the other variable.