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Let a and b belong to a commutative ring R. Prove that {x

Chapter 14, Problem 44E

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QUESTION:

Problem 44E

Let a and b belong to a commutative ring R. Prove that {x R | ax bR} is an ideal.

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QUESTION:

Problem 44E

Let a and b belong to a commutative ring R. Prove that {x R | ax bR} is an ideal.

ANSWER:

Step 1 of 2

We use Ideal Test to prove that  is an ideal of .

First take any two ; then we have , which means there are some  such that  and .

Now we compute the difference

because . This proves that .

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