This week, along with administrative matters, we covered basic definitions, examples, and types of rings, and the basic definition and some examples of ring homomorphisms. This material was all from Sections 2,3 and 4 of Tom’s notes.

Administrivia

We discussed general course policies, which can be found in the policies section of the webpage. Let me highlight office hours, which are currently scheduled for Tuesday 11-12 and Friday 10:30-11:30. Come to office hours! One of my regrets from Uni is not going to office hours enough.

Wednesday

Slides

On Wednesday we had a very brief quiz just to see what you remembered from MAS220. We then reviewed the formal definition of a ring, listed many examples, and introduced/reviewed three types of elements elements and the corresponding three types of rings:

  • a field is a ring in which every nonzero element is a unit
  • an integral domain is a ring which has no nonzero zero divisors
  • a ring is reduced if it has no nilpotent elements

    For each type of element/ring we found an example from our list of examples.

Thursday

Thursday we introduced the formal definition of a ring homomorphism, and gave some examples. We debated whether or not we should ask rings to contain multiplicative identities. We proved that there is a unique homomorphism from to any other ring . We defined a ring isomorphisms, and proved that a ring homomorphism is an isomorphism if and only if it is a bijection. We defined the kernel and the image of a homomorphism , foreshadowing our work for next week.