I have the following problem:
Let $R$ be a ring, and $M$ a monoid. we have the multiplication on $R[M]$ given by: $$\left(\sum_{m\in M} a_m\cdot m\right) \cdot \left(\sum_{m\in M} b_m\cdot m\right)=\sum_{m\in M}\left(\sum_{u\cdot v=m,\,\, u,v\in M} a_u\cdot b_v\right)m$$ I need to show that this multiplication is associative
I really have problems with the inizes when I multiply a third element to $\sum_{m\in M}\left(\sum_{u\cdot v=m,\,\, u,v\in M} a_u\cdot b_v\right)m$. Could maybe someone explain me how to work with the indizes here?
thank you very much.
The expressions make sense only if $M$ is finite or if we at least agree that only finitely many of the $a_m, b_m, c_m$ are non-zero. If you are overwhelmed by the indices, I suggest you try a proof by induction on e.g. the number of $m$ with $a_m\ne 0$.