i am doing the example constructed by BERGMAN in 1964 (see below for link), and i have a little doubt ,
he defines r.s=rs in Q(X) but Q(X) must have addition as operation as it is being checked for a module(add abelian group) and also the sufficient condition to check Q(X) is a right B module i.e. (r.Y).s=(r.α(s)).Y
please help.
http://www.ams.org/journals/proc/1964-015-03/S0002-9939-1964-0167497-4/S0002-9939-1964-0167497-4.pdf
The addition is the ordinary addition of $\Bbb Q(X)$.
The thing being constructed as $A$ is the twisted polynomial ring $D[Y;\alpha]=A$. It's got the same universal property that commutative polynomial rings have, so as soon as you determine the image of $D$ and $Y$, you've determined an entire homomorphism out of $A$. The rest of the images are decided using the distributive property.
By determining how $D$ and $Y$ act on $\Bbb Q(X)$, this creates a ring homomorphism $A\to End(\Bbb Q(X))$, and that is an $A$ module structure on $\Bbb Q(X)$.
