# rings and fields

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(See attached file for full problem description with proper symbols).

For part one....the first is in rational numbers, and second is in integers.

? Verify that is a sub field of and that is a sub ring of .

? Let R be a commutative ring and I an ideal of R. let and be elements of R. prove that is then

? Let M be an ideal of a commutative ring R and let with . Let . Prove that J is an ideal of R.

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