Exploring Monoid Rings in Aluffi's Algebra: Chapter 0

[Math Amateur]

I am reading Paolo Aluffi's book, Algebra: Chapter 0.

I have a question related to Aluffi's description of monoid rings ... ...

In Chapter III, Section 1.4 on monoid rings, we read the following:View attachment 4476
View attachment 4477

In the above text we find:"Given a monoid \(\displaystyle (M, \cdot )\) and a ring \(\displaystyle R\), we can obtain a new ring \(\displaystyle R[M]\) as follows.Elements of \(\displaystyle R[M]\) are formal linear combinations

\(\displaystyle \sum_{ m \in M} a_m \cdot m \)

where the 'coefficients' \(\displaystyle r_m\) are formal elements of \(\displaystyle R\) and \(\displaystyle a_m \ne 0\) for at most finitely many summands ... ... "
My question is as follows:What are the \(\displaystyle r_m\) exactly and what is their relation to the a_m ... ?Is it a typo ... that is does Aluffi mean .." ... ... where the 'coefficients' \(\displaystyle a_m\) are formal elements of \(\displaystyle R\) ... " ... that is ... does he mean \(\displaystyle a_m\) where he writes \(\displaystyle r_m\)?Hope someone can clarify this issue for me ... ...

Peter

 

[Fallen Angel]

Hi Peter,

I'm pretty sure that it's a typo, read it as $a_{m}$.

 

[Math Amateur]

Hi Peter,

I'm pretty sure that it's a typo, read it as $a_{m}$.

Thanks Fallen Angel ... certainly thought it might be ... but was not completely confident that I was not missing something ...

Thanks again,

Peter