Download Homework Assignment 5 MA407H Chapter 12 Chapter 13

Homework Assignment 5
MA407H
The sum of the maximum points is 100, and there are 10 extra credit points
Chapter 12
1. (6 points) The ring {0, 2, 4, 6, 8} under addition and multiplictaion modulo 10 has a unity. Find
it.
2. (8 points) Find an integer n that shows that the ring Zn need not have the following properties
that the ring of integers have.
(a) a2 = a implies a = 0 or a = 1.
(b) ab = 0 implies a = 0 or b = 0.
(c) ab = ac and a 6= 0 imply b = c.
3. (8 points) Describe the elements M2 (Z) that have multiplicative inverses.
4. (8 points) Suppose that R is a ring with unity 1 and a ∈ R such that a2 = 1. Let S := {ara | r ∈
R}. Prove that S is a subring of R. Does S contain 1?
Chapter 13
5. (12 points) Let R be ring of real valued function defined on all of R, with pointwise addition
and multiplication.
(a) Determine all zero-divisor of R.
(b) Determine all nilpotent elements in R.
(c) Show that every nonzero element of R is a zero-devisor or a unit.
√
√
6. (10 points) Show that Z7 [ 3] = {a + b 3 | a, b ∈ Z7 } is a field.
integer k and
√
√ For any positive
prime p, determine a necessary and sufficient condition for Zp [ k] = {a + b k | a, b ∈ Zp } to be a
field.
7. (12 points) Let x and y belong to a commutative ring R with charR = p 6= 0.
(a) Show that (x + y)p = xp + y p .
n
n
n
(b) Show that for all positive integers n, (x + y)p = xp + y p
(c) Find elements x and y is a ring of characteristic 4 such that (x + y)4 6= x4 + y 4 .
Ex. 8. (10 points) Consider the equation x2 − 5x + 6 = 0.
(a) How many solutions does this equation have in Z7 ?
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(b) Find all solutions of this equation in Z8 .
(c) Find all solutions of this equation in Z12 .
(d) Find all solutions of this equation in Z14 .
Chapter 14
9. (8 points) Find a subring in Z ⊕ Z that is not an ideal in Z ⊕ Z.
10. (8 points) Prove that I = h2 + 2ii is not a prime ideal in Z[i]. How many elements are in Z[i]/I?
What is the characteristic of Z[i]/I?
11. (10 points) Let Z2 [x] be the ring of all polynomials with coefficients in Z2 . Show that Z2 [x]/hx2 +
x + 1i is a field. How about Z3 [x]/hx2 + x + 1i?
divisor, so it cannot be a field.
12*. (extra credit 10 points) Show that Z[i]/h1 − ii is a field. How many elements does this field
have?
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