Say I have a ring of polynomials in $R[x]$. I wish to define the quotient group $R[x]/<x^2+1>$.
My question lies in the ideal generated by $<x^2 + 1>$. This is the set of all numbers such that if we take every element of $R$ it is
$I = r(x^2+1) $.
So it is every possible multiple of $x^2 + 1$. Is this correct, thus far?
Okay now, if this is our ideal, we wish to define a ring structure from creating additive cosets of this ideal. Is each element then from the quotient group described above equal to:
$R + I$
With the additive identity being equal to $0 + I$ and the Unity $1$ being equal to $ 1 + I$?
Is this all correct?