Expert: Sherry Wallin - 9/10/2010

Question
Dear Sir,

I am facing difficulties on the topics in Rings. Can you help me to solve the following question and I will like to know how to find the answers;

Find all the units in each of Z4, Z8, Z10 and Z14.

Thanks

Answer
Raymond~

Ask yourself what is a unit? By definition if R is a ring with unity, the elements in R that have multiplicative inverses are units. All Zn are rings with unity 1 so all you really want is to find the elements a in Zn for which a*b = e where b is the inverse of a and e = 1 of course.

Z4: 0,1,2,3
Is there a number times 0 that will give you 1? No of course not so 0 in NOT a unit

Is there a number times 1 that will give you 1? Well sure 1*1 = 1 so 1 is a unit.

Is there a number times 2 that will give you 1? Let's check it out:
0*2 = 0; 1*2 = 2; 2*2 = 0; 3*2 = 2 hmmm...there doesn't seem to be an inverse for 2 so 2 is not a unit

Is there a number times 3 that will give you 1? Let's check it out:
0*3 = 0; 1*3 = 3; 2*3 = 2; 3*3 = 1 aah! 3 is a unit since it has itself as an inverse (same as 1 above)

I think we can agree that 0 is never a unit in Zn, right? And 1 is always a unit in Zn, right?

I'll walk you through one more, I'll do Z10: 0,1,2,...,9
0 is not a unit; 1 is a unit; 2 is not a unit;

3 is a unit (7*3 = 21 = 1 mod 10)

Notice all I am doing is counting by multiples of the element and looking for a remainder of 1, so doing that let's check out 4 in Z10:
4,8,12,16,20,24,28,32,36 so 4 is not a unit

check out 5: 5,10,15,20,25,30,35,40,45 clearly 5 will never end with a remanider of 1

check out 6: 6,12,18,24,30,36,42,48,54 nope 6 isn't a unit either

check out 7: 7,14,21... 7 is a unit!

check out 8: 8,16, wait wouldn't you agree that a multiple of 8 or any even number has to end in an even number? So it will never have a remainder of 1 so 8 is not a unit

and finally check out 9: 9,18,27,36,45,54,63,72,81  9 is a unit!

Hopefully this has helped you understand what a unit is and how to find them.

Math Prof