Idea or Question  Example or Defintion  Extra Information 
1. What are the properties of a Ring?  
Example of infinite Commutive Ring with only one unity:  
Example of infinite Commutive Ring without unity:  
If a ring has a unity, or a ring element has a mulitplicative inverse, then ___________?  
5. S is a Subring of R if __________ ?  
Subring Test:  
Find all subrings of Z12:  
Find all subrings of Z12:  
Find all subrings of Z12:  
10. Find all subrings of Z12:  
Find all subrings of Z12:  
Definition of ZeroDivisors: (equation)  
Definition of Integral Domain:  
Cancellation:  
15. Zn is an Integral Domain when _________ ?  
Is QxQ an Integral Domain?  
Definition of Field:  
In a finite set what's the difference between integral domain and a field?  
Consequence of Questions 15 and 18.  
20. What are the elements of Z3[i]?  
Solve x^3 + 2 = 0 in Z3[i]. (Just solution set {a,b,c}  
If nx = 0 for all x in R, then n is __________  
The characteristic of R is ___________  
The characteristic of an ID is __________  
 Idea or Question  Example or Defintion  Extra Information 
25. If for all r in R, a in A, ra and ar are in A, then A is _________?  
Ideal Test:  
Let I = (a1, a2), then I is __________ ?  
R/A is a factor ring if and only if  
If ab in A implies a in A or b in A, then A is __________?  
30. If A is an Ideal with no proper Ideal containing it, then A is ___________?  
R/A is an ID if and only if  
R/A is an field if and only if  
Let f:R to S, with f(a+b) = f(a)+f(b), and f(ab) = f(a)f(b), then f is _________?  
If f is bijective it is __________?  
35. Under homomorphisms Kernels are ______?  
Let f:R to S be a ring homomorphism. Then R/Ker(f) is isomorphic to f(R), by r + Ker(f) > f(r).  
Let f: R > R/A, by f(r) = r + A. This shows ________  
If D[x] is not an integral domain, then  
In F[x], with functions f,g then there exists unique functions q,r such that f = gq + r, and _____________  
40. Let F be a field, a in F, f(x) in F[x]. Then f(a) is ____  
If f(a) = 0, then  
Polynomials of degree n have _______  
Let x^k  1 = 0. Then w = e^(2*pi*i/k) is called _______  
If every ideal has the form (a) = {ra  r in R}, the R is a _________  
45. Let F be a field, then F[x] is ___________  
I = (g(x)) if and only if ________  
x^2  4 / x + 2 in Z5[x] , Find q(x) and r(x)  
