Trying to understand quotient group subraction

Question

Say I have $R/I$ where, $$[a]=a+I$$

Then if $[a]=[b]$ does that mean $(a-b) + I=[0] $?

Answer 1

Yes, $[a]=[b]\iff a+I=b+I \iff a=b+i$ for some $i\in I$. Therefore, $a-b=i$ and $(a-b)+I=i+I=I=0+I=[0]$.