Ring theory problem

1. Jan 26, 2008

ehrenfest

[SOLVED] ring theory problem

1. The problem statement, all variables and given/known data
True or false. The product of two nonunits in Z_n may be a unit.

2. Relevant equations

3. The attempt at a solution
If a and b are two non units in Z_n, and ab <= n, then the result is clear, since ab would not be relatively prime to n. But what about if ab > n ?
Obviously the units form a group under multiplication, but I don't see how that helps.

2. Jan 26, 2008

ircdan

i don't see how it matters we are in Z_n,

if a and b are not units in a commutative ring R, then ab is not a unit, if it was then ..some stuff..

Last edited: Jan 26, 2008
3. Jan 26, 2008

ehrenfest

If ab were a unit, then we would have abc=1, for some c. This implies that a(bc)=1 which implies that a is a unit.

Is that it?

4. Jan 26, 2008

ircdan

of course yea! (just also notice by commutativity (bc)a = 1 too of course, but that's obvious)