Elements of $M\otimes_{\mathbb K}\mathbb K[t]$?

Question

I'm studying tensor product of modules over algebras and I came across the following statement.

Given a $\mathbb K$-module $M$, the elements of $M\otimes_{\mathbb K}\mathbb K[t]$ can be written as polynomials in $t$ with coefficients in $M$. I tried justifying this but I can't see why that holds.

Can anyone explain me this?

Thanks.

Answer 1

That's because $\,m \otimes \Bigl(\sum\limits_{i=0}^n a_i t^i\Bigr) $ can be written as $\,\sum\limits_{i=0}^n (a_im)\otimes t^i$, identified with $\,\sum\limits_{i=0}^n (a_im)t^i$.