1. Prove that for any positive integer n,
Ri of Q, i = 1, 2, . . . , n.
= cfw_ 0 for every choice of nonzero subrings
Solution: Suppose that R is a nonzero subring of Q. Then there exist positive integers r, s with r/s R,
and so r =
Noetherian Rings and Ane Algebraic Sets
Denition: A commutative ring R is said to be Noetherian or to satisfy
the ascending chain condition on ideals (or A.CC on ideals) if there is no
innite increasing chain of ideals in R, i.e., whenever I1 I2 I3 is
December 9, 2011
Let k be a eld, and A be a subset of An , the ane n-space over k .
I (A) = cfw_f k [x1 , . . . , xn ] | f (a1 , . . . , an ) = 0 for all (a1 , . . . , an ) A
For subset S of functions in the
Abstract: The presentation will prove the existence and uniqueness of
nite elds of order pn for every prime p and every positive integer n.
Denition : If F is a eld containing the eld K , then F is said to
be an extensionf ield of K . The mu
The Radical of an Ideal
Abstract: The radical of an ideal plays an important role in commutative
algebra, when we are concerned with the geometry aspects. This is due to the
bijection existing between varieties and radical ideals.
Mathematics 4123/9023 Assignment 3
1. Let R be a commutative ring with identity, and let S be a multiplicatively closed subset
of R containg neither 0 nor any zero divisors.
(a) Let I be an ideal of S 1 R.
(i ) Prove that I R = (we regard R to be a subrin
Mathematics 4123/9023 Assignment 2
1. Recall that for any nonempty set X and any ring R, the set of all functions from X to R
forms a ring with coordinatewise addition and multiplication, and we denoted this ring by
T = F (X, R).
(a) Prove that for any ch
Mathematics 4123/9023 Assignment 4
1. (a) Prove that x2 2 is irreducible in Z[ 2][x] (you may use the fact that Z[ 2] is a
Euclidean domain see Problem 9 in Section 8.1).
Solution: Since Z[ 2] is a Euclidean domain, it is a UFD and so we may apply Eisenst
1. (a) Let R be a ring, and let a R. Let Ra denote the intersection of all subrings of R
that contain a. Prove that Ra is a subring of R.
Solution: First, we note that a Ra , so Ra = . Let r, s Ra . Then for every ideal I of R that
Math 4123/9023 Final Exam
Due Monday, December 12
1. Let R be a ring with identity, and let M be a left R-module. Let N, K be R-submodules
of M such that K N . Let : M M/K denote the canonical surjective R-module
homomorphism, given by (m) = m + K for all
Without Light (%/s)
df = 12 and t = 4.9904
The P value equals 0.0003
The data is extremely significant as it is in the 99% range.
With Light (%/s)