Josiah Oh
Abstract Algebra
Assignment 1
12 January 2015
Let R be a ring with 1.
1. Show that (1)2 = 1 in R.
Proof. Since
(1)2 = (1)(1) = (1 1 1)(1) = 1 + (1)2 + (1)2 ,
after subtracting (1)2 and adding 1 to both sides, we obtain 1 = (1)2 .
11. Prove that
Josiah Oh
Abstract Algebra
Assignment 7
23 February 2015
Chapter 10.3: R is a ring with 1 and M is an R-module.
10.3.4 An R-module M is called a torsion module if for each m M there is a
nonzero element r R such that rm = 0, where r may depend on m (i.e.,
Josiah Oh
Abstract Algebra
Assignment 2
20 September 2015
Let R be a commutative ring with 1.
7.2.2 Let p(x) = an xn + an1 xn1 + + a1 x + a0 be an element of the polynomial
ring R[x]. We prove that p(x) is a zero divisor in R[x] if and only there is a
non
Josiah Oh
Abstract Algebra
Assignment 8
2 March 2015
12.1.2 Let M be a module over the integral domain R.
(a) Suppose that M has rank n and that x1 , x2 , . . . , xn is any maximal set
of linearly independent elements of M . Let N = Rx1 + + Rxn be
the sub
Josiah Oh
Abstract Algebra
Assignment 6
16 February 2015
9.4.1 Determine whether the following polynomials are irreducible in the rings indicated. For those that are reducible, determine their factorization into irreducibles. The notation Fp denotes the n
Josiah Oh
Abstract Algebra
Assignment 5
9 February 2015
8.3.8 Let be a the quadratic integer ring Z[ 5]. Prove that 2, 3, 1 + 5 and
R
1 5 are irreducibles in R, two of which are associate in R, and that
no
6 = 2 3 = (1 + 5) (1 5) are two distinct factoriz
Josiah Oh
Abstract Algebra
Assignment 3
26 January 2015
7.4.7 Let R be a commutative ring with 1. Prove that the principal ideal generated
by x in the polynomial ring R[x] is a prime ideal if and only if R is an integral
domain. Prove that (x) is a maxima
Josiah Oh
Abstract Algebra
Assignment 4
2 February 2015
8.2.3 Prove that the quotient of a P.I.D. by a prime ideal is again a P.I.D.
Proof. Let R be a P.I.D. and let I be a prime ideal. Then I is a maximal ideal
by a proposition. Then R/I is a eld by anot