Study Guide for Ring and Field Theory

Here are some definitions for the upcoming Ring and Field Theory midterm.

• Ring: A ring R is a set with two binary operations -- generally called addition and multiplication -- such that, for all a, b, c in R, the following properties hold:
  

1. a+b=b+a (the ring is commutative under addition)
2. there exists an additive identity element called 0 such that 0+a=a+0=a
3. there exists an additive inverse of each element such that a+(-a)=0
4. a+(b+c)=(a+b)+c (the ring is associative under addition)
5. (ab)c=a(bc) (the ring is associative under multiplication)
6. a(b+c)=ab+ac and (b+c)a=ba+ca (multiplication distributes over addition.
   Examples: Z; nZ; M2(Z), or the ring of 2x2 matrices with integer entries; Z[x], ring of polynomials with integer coefficients

• Zero-divisor: In a ring R, an element a is a zero-divisor if there exists an element b such that ab=0.
  Examples: in Z(4), 2 is a zero-divisor.
  
• Integral domain: An integral domain is a ring R such that R has unity (a multiplicative inverse), R is commutative, and there are no zero-divisors.
  Examples: Z; M(2)Z; Z(p), where p is prime; NOT Z(n), where n is NOT prime.
  
• Field: A field F is a ring such that F has unity, F is commutative, and every nonzero element in F has a multiplicative inverse.
  Examples: R; Q; any finite integral domain; Z3(i).
  
• Characteristic of a ring: The characteristic of a ring R is the smallest positive integer n such that, for all a in R, na=0. If no such positive integer exists, then the characteristic is 0.
  Examples: In Z(n), the characteristic is n; in Z, the characteristic is 0; in Z[x], the characteristic ic 0.
  
• Ideal: An ideal I is a subring of a ring R such that, for all a in I and for all r in R, ar and ra are in I.
  Examples: In Z, nZ is an ideal; in Z[x], nZ[x] is an idea; in a commutative ring R with 1 and a, the set , where 'a'={ra| r in R} is called the principle ideal generated by a; similarly, the set 'a,b'=(r1a+r2b| r1, r2 in R}, and so on.
  
• Prime ideal: An ideal I is prime if it is a proper ideal and if ab in I implies that a is in I or b is in I.
  Examples: In Z, nZ is a prime ideal iff n is prime.
  
• Maximal idea: A proper ideal I is maximal to R if, given an ideal I', I subsets I' implies that I=I' or that I'=R.
  Examples: In Z, nZ is maximal iff n is prime; is maximal in Z[x].
  
• Ring Homomorphism: A ring homomorphism P is a mapping from a ring R to a ring S such that, for all a,b in R, P(a+b)=P(a)+P(b) and P(ab)=P(a)P(b), that is, if P preserves both operations.
  Examples: if P maps all elements in R to 0, then P is a homomorphism; there is no homomorphism from 2Z to 3Z (or to 4Z); for any pos int n, the mapping from k to kmodn is a ring homomorphism.
  

Proofs to come soon.
Bubye.