# Ring Isomorphism

• Oct 31st 2008, 05:57 AM
vincisonfire
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).
• Oct 31st 2008, 10:53 AM
ThePerfectHacker
Quote:

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 $\displaystyle R/I$ are of the form $\displaystyle aI$. Similarly elements in $\displaystyle (R\times S)/(I\times J)$ are of the form $\displaystyle (a,b)(I\times J)$.

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