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Math Help - 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 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.
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