Proofs and Theorems for Exam

Here are some proofs that I may need iterate on the exam. Following that are some theorems that I'll probably need to know.

• Claim: A finite integral domain is a field.
  Pf: Let D be a finite integral domain with unity 1. Let a be any nonzero element in D. I will show that a is a unit. If a=1, it is its own inverse, so I am done. So, assume that a =! 1. Then, let D contain a, a^2, a^3, a^4.... D is finite by assumption, so there must be two positive integers i and j such that i>j and a^i=a^j. Then, a^(i-j)*a^(j)=1*a^j. By cancellation, a^(i-j)=1. Since a=!1, i-j>1. Then, a*a^(i-j-1)=1. So, a^(i-j-1) is the inverse of a. qed.
  
• Claim: The characteristic of an integral domain is 0 or prime.
  Pf: It suffices to show that if the additive order of 1 is finite, it must be prime, because, in a ring with unity, the additive order of 1 is characteristic of the ring. Suppose that 1 has order n and that n=st, where 1<=s, t<=n. Then, 0=n*1=(st)*1=(s*1)*(t*1). There are no zero-divisors in an integral domain, so either s*1=0 or t*1=0. Since n is the least positive integer with the property that n*1=0, we must have s=n or t=n. Thus, n is prime. qed.
  
• Let R be a commutative ring with unity and let A be an ideal of R.
  Claim: R/A is an integral domain iff A is prime.
  Pf: Suppose that R/A is an integral domain and ab in A. Then, (a+A)(b+A)=ab+A=A. So, either a+A=A or b+A=A; that is, either a in A or b in A. Then, A is prime. To prove the converse, observe that R/A is a commutative ring with unity for any proper ideal A. Suppose that A is prime and that (a+A)(b+A)=0+A=A. Then, ab in A. Then, since A is prime, a in A or b in A. Then, a+A=A or b+A=A. That is, there can be no zero-divisors. qed.
  

There are two more that I am encouraged to know, but I don't think I'll need to know proofs. He generally only asks for proof of a theorem found in the book, but gives us three or four different options and lists five different theorems on the study guide. So, I'll be fine if I know these three.
Here are some theorems that I'll probably need to know:

• Cancellation: Let a, b, c be in an integral domain. If a=!0 and ab=ac, then b=c. NOTE: THIS IS NOT DIVISION.
  
• A finite integral domain is a field.
  
• For every prime p, Z(p) is a field.
  
• Let R be a ring with unity 1. If 1 has infinite additive order, then the characteristic of R is 0. Else, if 1 has order n, then the characteristic of R is n.
  
• The characteristic of an integral domain is 0 or prime.
  
• R/A is an integral domain iff A is prime.
  
• R/A is a field iff A is maximal.
  
• The kernel of any homomorphism is an ideal the ring.
  
• Every ideal A of a ring R is the kernel of a ring homomorphism r-->r+A from R to R/A.
  
• A ring with unity contains Z(n) or Z.
  
• A field contains Z(p) (if char=n) or Q (if char=0).
  

Ok, that's about all I need to know. Not bad, really. Its just the first midterm. Not bad at all.