Let R and S be rings and let I ▹R, J ▹S be ideals. Are the elements of (RxS), (IxJ) and (R×S)/(I×J) [quotient ring] of the form (a,b) just like two components vectors?

(With more rigorous thoughts) It could lead to (R×S)/(I×J) = (R,S)/(I,J) = (R/I , S/J) = (R/I) x (S/J).