Solved: False - Product of Two Nonunits in Z_n Cannot be Unit

[ehrenfest]

[SOLVED] ring theory problem

Homework Statement

True or false. The product of two nonunits in Z_n may be a unit.

Homework Equations

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.

 

[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..

 

[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?

 

[ircdan]

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