A quotient ring, if anything, is the quotient group of the abelian group underlying the ring, right? So, in particular by definition (the index), but I think you'll find that .
what you know is that the coset a+bi + J is the same as a(mod 3) + b(mod 3)i + J. that is a,b = 0,1 or 2 (mod 3). this means we get exactly 9 (distinct) elements:
what you know is that the coset a+bi + J is the same as a(mod 3) + b(mod 3)i + J. that is a,b = 0,1 or 2 (mod 3). this means we get exactly 9 (distinct) elements: