Name:
Problem Set 8
Math 415 Honors, Fall 2014
Due: Tuesday, November 18.
Review Sections 18, 19 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Tuesday, November 18. Remember to fully justify all your answers,
and provide complete details. Neatness is greatly appreciated.
1. Read Example 18.10 in the text. Show that φa satisfies the properties of the definition of ring
homomorphism.
2. Let R = M2 (Z2 ) be the set of all 2 × 2 matrices with coefficients in Z2 . Show that R is a ring
with usual addition and multiplication, find the unity, find the number of elements of R, and
list all units in R.
√
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3. Show that the sets R1 = {a + b 2 | a, b ∈ Q} and R2 = {a + b 2 | a, b ∈ Z} are rings with usual
addition and multiplication. What kind of rings are they? (commutative? with unity? integral
domain? division ring? field?)
4. Describe all ring homomorphisms of Z into Z and all ring homomorphisms of Z × Z into Z.
5. Let R be a ring with unity and let U be the set of all units in R. Show that U is a group with
the operation of multiplication in the ring.
6. Show that the fields Q and R are not isomorphic. Show that the rings 6Z and 7Z are not
isomorphic.
7. Let p be a prime number. Show that, for a, b ∈ Z p , (a + b) p = a p + b p (in Z p ).
8. Show that if r, s are positive integers and gcd(r, s) = 1, the rings Zr ×Z s and Zrs are isomorphic.
Use this result to prove that for m, n ∈ Z, there exists an integer x such that x ≡ m mod r and
x ≡ n mod s.
9. Let R be a commutative ring with unity of characteristic 3. Let a, b ∈ R. Find a simplified
expression for (a + b)6 and for (a + b)9 .
10. Show that a finite ring with unity 1 , 0 and no zerodivisors is a division ring.
11. Find the characteristic of the rings Z21 × Z35 , Z5 × 5Z, and Z × Q.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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