Study guide for first exam
You should know:
Definitions and examples of: Ring (with unity), integral domain, field,
characteristic of a ring, unit,
nilpotent element, idempotent, zero divisor, ideal,
quotient/factor ring, homomorphism,
isomorphism. Examples of rings: Z,
Q, R, C,
Zm, R[x], Z[i], Q[sqrt 2], etc. Prime and maximal
ideals.
Theorems:
- The characteristic of an integral domain is 0 or a prime
- R/I is an integral domain if and only if I is prime
- R/I is a field if and only if I is maximal
- I maximal ==> I prime
- The first isomorphism theorem
- Lots of properties of homomorphisms (Theorem 15.1)
- The kernel is an ideal, and all ideals are kernels
Some study questions:
These are just some suggestions of problems to give you more practice.
Chapter 12: 24, 34, 37
Chapter 13: 12, 15, 21, 34, 38
Chapter 14: 29, 34, 41, 47, 51
Chapter 15: 26, 37 (you can leave out the "onto" part, since our
homomorphisms take unity to unity)
Changes to the text
Here are the places I have found in the text we have covered where
requiring rings to have
a unity changes things.
Chapter 12:
Skip examples 5, 9, 10. In
Theorem 12.3(subring test), you must also check 1 is in S.
Chapter 13:
Theorem 13.3 can be taken as the
definition of characteristic. Note that although my definition of
ideal
looks different, it is equivalent to the one in the book.
Chapter 14:
Skip example 7, 9
Chapter 15:
A ring homomorphism R --> S must
take the unity 1R to the unity 1S. Skip
examples 4, 5, 7.