Motivated by the image and kernel of a ring homomorphism, this week we introduced the notion of subring and idea; we discussed subrings and Wednesday and ideals on Thursday.
This material was all from Sections 5 and 6 of Tom’s notes.
On Wednesday we briefly ran over exercise 3 from the homework, and then discussed subrings. The one potentially tricky part of the definition is that the identity of the subring has to be the identity of the big ring.
We then discussed generating subrings, after some motivation of generating groups; the subring generated by a set is denoted ; intuitively, this should be mashing up everything in in all possibly ways with addition and multiplication. That’s a bit messy, so the formal definition is taken to be the intersection of all subrings containing . We discussed rings that are finitely generated, and rings that are not finitely generated.
Thursday we discussed the results of the first problem set a bit at beginning. We then discussed ideals of rings, very much in parallel to the discussion of subrings on Wednesday. The ideal generated by a finite set is denoted , and again had an elegant and impractical definition as the intersection of all ideals containing , and a hands on version as -linear combination of elements of .
We gave an analogy of generating rings as being like taking polynomials, while generating ideals is like generating a vector space; this analogy will be developed further.
We proved that every ideal of the integers is generated by one element; such rings are called principal ideal domains; we gave examples and non-examples of P.I.D.s