Midterm Exam
1. Prove that for any positive integer n,
Ri of Q, i = 1, 2, . . . , n.
n
i=1 Ri
= cfw_ 0 for every choice of nonzero subrings
Solution: Suppose that R is a nonzero subring of Q. Then th
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 increasi
Hilberts Nullstellensatz
Barum Rho
December 9, 2011
1
Motivation
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 al
Finite Fields
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
Majed Albaity
09/12/2011
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
b
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
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 denot
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 Eu
Assignment 1
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 = . Le
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 surject
Photosynthesis Rates
Without Light (%/s)
-0.0002903
-5.69x10-5
-0.0001259
-0.0001126
-0.0010191
-0.0006336
5.7211x10-6
t-test:
df = 12 and t = 4.9904
The P value equals 0.0003
The data is extremely si