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# Ideals watch

1. I am trying to understand the concept of Ideals in ring theory, and so far I think I get it but would like some confirmation.

is the ideal generated by

(By the way I know there exist left and right ideals but I do not need to know this as I am working with commutative rings).

I have:

etc for ideals with more elements.

There is a line in my notes which is:

"Sometimes you don't need two elements: "

I understand that this works, but what I do not get is how you can tell whether two ideals are equal.

Is there a way to tell that two ideals are equal or is it just inspection.

I also think the way I have written the line in my notes is wrong. I mean I understand that it means:

Both and are ideals for the ring and that , but is there a way in which to write this without saying it in words.

2. Let's use the (15,21) and (3) example.

Now, the hcf of 15 and 21 is 3, and so there exist integers a and b such that ; so in particular, . But then notice that , so any multiple of 3 is in the ideal, and so .

Also, if then for some , and so . In other words, . But by the previous result, this means that .

In general, we can define "highest common factors" in commutative rings. We say that is a highest common factor of a and b if and if and then . These can be useful for checking if two rings are equal, but not always (since we can't always write d=ax+by).
3. (Original post by nuodai)
In general, we can define "highest common factors" in commutative rings. We say that is a highest common factor of a and b if and if and then . These can be useful for checking if two rings are equal, but not always (since we can't always write d=ax+by).
You can define them, but I think it's worth pointing out that they do not have to exist.
4. Thanks

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