Math 366–002 HW 6, Spring 2014

Math 366–002 HW 6, Spring 2014
This assignment is due Friday, April (!) 4 in class. Feel free to work together, but be
sure to write up your own solutions.
As for writing it up, please write legibly on your own paper, including as much justification
as seems necessary to get the point across.
1. What is the unity element of 2Z10 = {0, 2, 4, 6, 8} ⊂ Z10 ?
2. Suppose R is a ring that is cyclic under addition. Show that R is a commutative ring.
3. Show that a unit a of a ring R divides every element of R, i.e., ∀b ∈ R, b = ac for some
c ∈ R.
4. Show that 2Z ∪ 3Z is not a subring of Z.
5. Write down the list of zero divisors in Z12 and the list of units in Z12 .
6. Show that every nonzero element of Zn is a unit or a zero divisor. In particular, suppose
a ∈ Zn is not a zero divisor and show that it must be a unit.
7. Let R be a ring with unit, and suppose a is nilpotent, i.e., there is some positive
integer n such that an = 0. Show that 1 − a ∈ R is a unit. (Think about the product
(1 − a)(1 + a + . . . + an−1 ).)
8. Find the multiplicative inverse of 4x2 + 6x + 3 in Z8 [x].