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Math Help - ring

  1. #1
    Senior Member Sampras's Avatar
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    ring

    Let  I and  K be ideals in a ring  R with  J \subseteq I . Prove that  I/K = \{a+K: a \in I \} is an ideal in the quotient ring  R/K .

    Closure under subtraction follows from the fact that  I is an ideal.

    Suppose  r+K \in R/K and  a+K \in I/K . Then  (r+K)(a+K) = ra+K \in I/K since  I is an ideal. Likewise,  (a+K)(r+K) = ar+K \in I/K . So the result follows. Is this correct?
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  2. #2
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    Quote Originally Posted by Sampras View Post
    Let  I and  K be ideals in a ring  R with  J \subseteq I . Prove that  I/K = \{a+K: a \in I \} is an ideal in the quotient ring  R/K .

    Closure under subtraction follows from the fact that  I is an ideal.

    Suppose  r+K \in R/K and  a+K \in I/K . Then  (r+K)(a+K) = ra+K \in I/K since  I is an ideal. Likewise,  (a+K)(r+K) = ar+K \in I/K . So the result follows. Is this correct?

    Yes, it seems it is. But that J above must be K...right?

    Tonio
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