| Idea or Question | Example or Defintion |
|---|
| 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 Zero-Divisors: (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 |
|---|
| 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) | |
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