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September 16th, 2012, 04:02 PM   #1
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ideal

hey

lets say that (a, b) is an ideal in R. What exactly is that set? How can i describe an element of (a, b)?

I would like to know this, because i have to show that some kind of set is equivalent to an ideal (generated by 2 elements) and i haven't got a clue how to do that :-s

help is appreciated very very very much!
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September 16th, 2012, 09:04 PM   #2
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Re: ideal

The ideal is the same as the set . It is the set of all sums of elements for any . Note that , so elements of the form and are in , hence and .
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September 17th, 2012, 07:03 PM   #3
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Re: ideal

Thank you!

But what is then the difference between the ideal (a, b) and the sum of the ideals (a) and (b)?

And if we multiply the ideal with itself? Is it then the same ideal?
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September 20th, 2012, 06:47 PM   #4
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Re: ideal

There is no difference at all between and , they are the same sets.

The product is the set of finite sums where and , so the product is actually given by finite sums , which is to say that . So in order that , it must be that which is unlikely, but not impossible depending on the ring.
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