What is the ideal generated by a and b.

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The ideal generated by elements a and b in a ring R, denoted as (a, b), is defined as the set of all sums of the form ra + sb, where r and s are elements of R. This is distinct from the union of the ideals aR and bR, which does not capture the closure properties required for an ideal. The notation (a, b) does not refer to ordered pairs but rather to the smallest ideal containing both a and b, aligning with the general definition of generated objects in algebra.

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what is "the ideal generated by a and b."

I'm doing a problem and I'm a little confused on the notation.

Let R be a ring and a,b \in R. Prove that (a, b), the ideal generated by a and b is the...

I want to do this on my own so I left out the rest of the problem. I just need to know if (a, b) = aR \cup bR. Or is it some kind of set of ordered pairs?
 
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It's not the union aR and bR. It's the set of all sums of the form aR+bR. They aren't the same. And unless (a,b) is some funny notation different from {a,b} then it has nothing to do with ordered pairs.
 


Okay that makes sense. The way I described it it wouldn't even be closed under addition.
 


General remark: Usually the phrase "object of type T generated by a set X" means "smallest object of type T containing the set X." And usually this is the 'intersection' (or appropriate analogue) of all objects of type T containing the set X.
 

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