What is the ideal generated by a and b.

[futurebird]

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?

 

[Dick]

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.

 

[futurebird]

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

 

[morphism]

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.