The definition of the crossed product of a ring and a monoid I know the  definition of crossed product of a ring and a group. But when I consider the definition of the crossed product of a ring and a monoid, I can not understand exactly  the definition on page 27 of the book "Ring Constructions and Applications" by Andrei V. Kelarev. I want to know if the set $\bar{S}$ can be replaced directly by the monoid $S$, and if this replacement seems to make sense? that is to say, we only need to consider the finite sums of each term of R*S as the form of  $r_{s}s$? Any help would be helpful. Thank you very much!
 A: Fast answers
You can write elements of $M$ without bars in the crossed product, just as long as you don't forget that they don't multiply according to the rules of the monoid $M$.
Yes, you only take finite sums of the $r_s s$: there's no reason to expect infinite sums are defined. The ring is, by definition, the free module on the elements $\{\overline m\mid m\in M\}$, and that can be expressed by finite linear combinations only.
Elaboration
The crossed product $(R,M,\tau,\sigma)$ is a construction generalizing both skew monoid rings and twisted monoid rings.
Twisted monoid rings
In the usual monoid ring $R[M]$, $M$ injects directly in as a submonoid, and its monid multiplication matches the multiplication in $R[M]$. But in a twisted monoid ring $(R,M,\tau)$, the set $M$ still injects into the twisted ring, but $M$ is not usually a submonoid of $R$ because its product doesn't match. 
That is why the overlines are used for the images of $M$ in $(R,M,\tau)$: they don't want you to look at $a,b\in M$ and say "ah, well if I multiply these inside the twisted group ring I get $ab$." The overlines are added to remind you that they are multiplied differently: $\overline{a}\overline{b}=\tau(a,b)\overline{ab}$. 
(By the way, there is either a typo in the googlebooks preview I'm reading, or else the resolution is bad, because it looks like $\overline{ab}=\tau(a,b)\overline{ab}$, but that doesn't make sense. See page 13 of Passman's Algebraic structure of group rings instead.)
Skew monoid rings
A skew monoid ring $(R,M,\sigma)$ generalizes a monoid ring again by not assuming that the elements of $R$ commute with the elements of $M$: that is, $mr=\sigma(r)m$ where $\sigma$ is a ring homomorphism of $R$ into itself. In this skew ring, you can still write elements of $M$ without bars because they are multiplied in $(R,M,\sigma)$ and in $M$ exactly the same way.
Crossed products
This is what you get when you do twisting and skewing simultaneously. It's inadvisable to drop the bars from the elements of $M$ because of the twisting.
