# Math Help - Ring Isomorphism

1. ## Ring Isomorphism

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).

2. Originally Posted by vincisonfire
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).
Elements in $R/I$ are of the form $aI$. Similarly elements in $(R\times S)/(I\times J)$ are of the form $(a,b)(I\times J)$.

While to prove,
$(R\times S)/(I\times J) \simeq (R/S) \times (I/J)$
You would have to find a ring homomorphism from $R\times S$ to $(R/S)\times (I/J)$ that is onto and has kernel $I\times J$ then procede to invoke the fundamental homomorphism theorem.