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Thread: Ring Isomorphism

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    Senior Member vincisonfire's Avatar
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    Ring Isomorphism

    Let R and S be rings and let I ▹R, J ▹S be ideals. Are the elements of (RxS), (IxJ) and (RS)/(IJ) [quotient ring] of the form (a,b) just like two components vectors?
    (With more rigorous thoughts) It could lead to (RS)/(IJ) = (R,S)/(I,J) = (R/I , S/J) = (R/I) x (S/J).
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    Quote Originally Posted by vincisonfire View Post
    Let R and S be rings and let I ▹R, J ▹S be ideals. Are the elements of (RxS), (IxJ) and (RS)/(IJ) [quotient ring] of the form (a,b) just like two components vectors?
    (With more rigorous thoughts) It could lead to (RS)/(IJ) = (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.
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