MAT 412 HOMEWORK 4
DUE: FEBRUARY 3, 2017 (BEGINNING OF CLASS)
This homework set covers section 2.2, 2.3 and 3.1. References are to Hungerford, 3rd.
edition.
Problem 1. (a) Write out the addition and multiplication table of Z4and Z6.
(b) Write out the addition and multiplication table of Z2×Z2and Z2×Z3(Here addition
and multiplication is defined componentwise, i.e., ([a],[b]) ⊕([c],[d]) = ([a]⊕[c],[b]⊕[d])
and ([a],[b]) ([c],[d]) = ([a][c],[b][d]))
(c) If you compare the tables for Z4and Z2×Z2, are they the “same”? Same question for
Z6and Z2×Z3.
Problem 2. (2.2.B.14) Solve the following equations:
(a) x2+x= [0]in Z3
(b) x2+x= [0]in Z6.
(c) If pis prime, show that the only solutions of x2+x= [0]in Zpare [0]and [p−1].
Problem 3. (a) Find all the units and zero-divisors in Z8and Z19.
(b) How many units are there in Zp, where pis prime?
(c) Do all elements of Z8have square-roots? (An element [a]in Znis a square-root if
there exists a [b]∈Znsuch that [a][a]=[b]).
Problem 4. Show that every nonzero element of Znis either a unit or a zero divisor, but
not both. (2.3.B.10)
Problem 5. (a) Show that R2together with componentwise addition and multiplication
∗defined by
(a,b)∗(c,d) = (ac −bd,ad +bc)
is a commutative ring. We will write R0for this ring, that is, R0is the ring (R2,+,∗).
[To find the multiplicative unit 1R0it might be useful to find a solution for the equation
(a, 0)∗(x,y) = (a, 0)]
(b) Show that in R0, the equation x2+1R0=0R0has a solution. [Remark: In fact, one can
show that every polynomial equation has a solution in R0]
(c) The ring R0is better known under which name?
Problem 6. Let M2(R)be the set of all 2 ×2-matrices with real entries. Define GL2(R) =
{A∈M2(R): there exists B∈M2(R), such that AB =BA =12}.
(a) Is GL2(R)a ring with matrix addition and matrix multiplication? If not, which prop-
erties fail?
(b) Let D:={A∈M2(R):Ais a diagonal matrix, i.e., A=α0
0α}. Is Da ring with
matrix addition and matrix multiplication? Are there zero-divisors in D? (A zero-divisor
in Dwould be an A6=0D∈Dsuch that there exists a B∈Dwith AB =0D).
Problem 7. Read ahead sections 3.2 and 3.3 in the book.