I'm taking a modern algebra course and having some trouble understanding how ideals are generated. My book (Dummit and Foote) says that for a ring R and a subset A, if we let (A) be the smallest ideal containing A, (A) is the ideal generated by A. They then go on to give an explanation about how (A) is the intersection of all ideals in R containing A.
This is all well and I guess I don't have any trouble with that on the surface, but I'm still not really sure what an ideal is or how to use one practically. I'm trying to go further in the book and do some stuff with field theory, and I find them talking about (p(x)), the ideal generated by a polynomial p(x) (in Z[x], for example).
My only understanding of that right now is that (p(x)) is the set of things of the form q(x)*p(x), with q(x) an element of Z[x]. I also know that if you multiply anything in the ring by something in the ideal, the product should stay in the ideal too, so that would be consistent with this.
Is this right? Is there anything else major I should know?