Let $R$ be a commutative ring with 1 and $I$ the ideal generated by $R$. I have:
$R/I=\{[a]_I:a\in R\}$ where I can define its sum as: $[a]_I+[b]_I=[a+b]_I$. For $X,Y \in P(R)$ (power set) I define the complex sum as: $X+Y=\{x+y|x \in X, y \in Y\}$.
Are my definitions correct?
Take any $c\in [a+b]_I$. Then $c-(a+b) \in I$ by the definition of a congruenct class modulo $I$. We have $c=a+b+[c-(a+b)]$. I want to show that $c$ is the sum of two elements of subsets of $R$.
$c-(a+b)$ is in $I$ which is a subset of $R$ (?). And $a+b$ is in $R$?
I haven't started with the opposite direction yet. Thanks for any help!