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Can you name the Properties of Rings?

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Idea or QuestionExample or DefintionExtra 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 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 QuestionExample or DefintionExtra 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)

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