I an going through a class in abstract algebra, as we go from one chapter to another, the quotient theory comes back over and over, but the teacher never explicitly states the equivalence relations used, I am rather unsure how is the relation equivalence defined when we work, for example, with: $$\mathbb{R}[X]/(X^2 + 1)$$
If I take two polynomials $A,B \in \mathbb{R}[X]$, then we have $A = (X^2 + 1)Q + R$ and $B=(X^2 + 1)Q' + R'$, so what is the condition so that $A \sim B$? Would it be $R = R'$, or maybe $Q = Q'$.
The easiest way to say it is that $P \sim Q$ if $P-Q \in (X^2+1)$.
To see this, suppose that $P(x) \sim Q(x)$, then $P(x)-Q(x)= g(x)(x^2+1)$, so $P(x)=Q(x)+g(x)(x^2+1)$, or in other words, $P(x) \in Q(x)+(x^2+1)$, and likewise, $Q \in P(x)+(x^2+1).$
The same can be said when they are equivalent in the quotient.