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Math Help - If A is a ring, prove that A-module ,A[x] is isomorphic with directs sum of Ai....

  1. #1
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    If A is a ring, prove that A-module ,A[x] is isomorphic with directs sum of Ai....

    Is true this statements ??
    a) If A is a ring, prove that A-module, A[x] is isomorphic with directs sum of Ai. were Ai=A , for i=1,2,3...


    , $A[x] \simeq \Sigma_{i \epsilon N}A_i$  how A-module.
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    Re: If A is a ring, prove that A-module ,A[x] is isomorphic with directs sum of Ai..

    Quote Originally Posted by john27 View Post
    Is true this statements ??
    a) If A is a ring, prove that A-module, A[x] is isomorphic with directs sum of Ai. were Ai=A , for i=1,2,3...


    , $A[x] \simeq \Sigma_{i \epsilon N}A_i$  how A-module.
    A module M over an underlying ring A is said to be free if it has a A-basis. Can you write down a basis for A[x]? Can you see why a free A-module is necessarily a direct sum of copies of A?

    Hint: Look at the definition of direct sums and see how it corresponds to the definition of a basis.

    As an aside, we usually write \oplus_{i\in \mathbb{N}} A_i instead of \sum_{i\in \mathbb{N}}A_i
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    Re: If A is a ring, prove that A-module ,A[x] is isomorphic with directs sum of Ai..

    yes or no ?

    Are this A-modules isomorphic :  $A[x] \simeq \oplus _{i \epsilon N} A$ ??
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