Change of base rings for exterior algebra

Question

This may be not a good question. But I really get tough. I am studying basic knowledge about homological algebras and I am dealing with Koszul's Complex and Hilbert's Syzygy Theorem. At the very beginning, I need to know some properties about exterior algebras. There is a lemma in N.Jacobson's Basic Algebra(2nd)(Please see the following picture).
The change of base rings is mentioned above and I consider that this lemma wants to tell us that if we change the ring acting on the module $M$ as above, we will have $E(M)_L\cong E(M_L)$ . But I am confused about the notations of $E(M)_L$ and $E(M_L)$.

In my opinion, I regard $E(M)_L= (K_L\oplus M_L \oplus (M\otimes_KM)_L\oplus \,...)/I$, where $I$ is the ideal generated by $m^2$, $m\in M$. And I think $E(M_L)=(L\oplus M_L \oplus (M\otimes_LM)\oplus \,...)/I$, where $I$ is the ideal generated by $m^2$, $m\in M$. But when I think like this, the proof seems to strange for me. The first sentence said that $M_L$ can be identified with the subset of $E(M)_L$. And elements in this subset are of the form $\sum l_i\otimes x_i$. My question is if what I think above is right, why will the subset of $E(M)_L$ have that form? it is just the form to show the action of $L$ on $M$?
So I want to ask what do these two notations mean. And if there is another proof of this lemma, it is really kind of you to tell me where it is.
Thanks for your assistance!

Answer 1

Tensor product is not exact, it is only right exact. But the tensor product does take split exact sequences to split exact sequences, i.e. tensoring $$M \to M \oplus N \to N$$ with $T$ gives $$T \otimes M \to (T \otimes M) \oplus (T \otimes N) \to T \otimes N$$ That means that if you have the inclusion of a direct summand $M \hookrightarrow E$ then tensoring with a module yields an inclusion $T \otimes M \hookrightarrow T \otimes E$.

So take the inclusion $M \hookrightarrow E(M)$ and tensor with $L$ to get $M_L \hookrightarrow E(M)_L$. The image of $M_L$ under this map will be the elements in $E(M)_L = L \otimes_K E(M)$ generated by the simple tensors $\ell \otimes m$ where $m \in E(M)$ is an element of $M$, i.e., $m$ is in the image of $M \hookrightarrow E(M)$.