Thread: Group of Units

Show that the set of units, U, of R forms a group under multiplication.

My thoughts... U is a group since if we multiply a unit by a unit we get a unit. (not sure if true and would probably need a proof...)
Formally, let a,b be units. Then a|1 and b|1. So ab|1, hence ab is a unit. So is a group under multiplication...

Just a draft of what i think the answer is. Pretty sure there's more to it though. Am i on the right track?

Originally Posted by Deadstar

Show that the set of units, U, of R forms a group under multiplication.

My thoughts... U is a group since if we multiply a unit by a unit we get a unit. (not sure if true and would probably need a proof...)
Formally, let a,b be units. Then a|1 and b|1. So ab|1, hence ab is a unit. So is a group under multiplication...

Just a draft of what i think the answer is. Pretty sure there's more to it though. Am i on the right track?

Hi Deadstar.

1 is clearly a unit since

If and are units, then and for some Hence showing that is a unit.

And the above also shows that the multiplicative inverse of is Hence the set of all units of a ring with unity forms a group under multiplication.

Notice that I have been careful not to assume that is a commutative ring since you did not state whether it is or not.