Basic proof of units in a ring with identity.

[shamus390]

Homework Statement

Let R be a ring with identity, and a,b are elements in R. If ab is a unit, and neither a nor b is a zero divisor, prove a and b are units.

Homework Equations

If ab is a unit then (ab)c=1=c(ab) for some c in R.

The Attempt at a Solution

Assume both a and b are not zero divisors, and denote the identity element of R as 1.

Since ab is a unit in R, there exists some c such that (ab)c = 1.
(ab)c=1 \Rightarrow (ca)b = 1 \Rightarrow b is a unit.

Similarly, (ab)c= 1 \Rightarrow a(bc) \Rightarrow a is a unit.

Does the above sufficiently prove the claim in the problem statement? I feel like I'm missing something.

Thanks!

 

[Dick]

Homework Statement

Let R be a ring with identity, and a,b are elements in R. If ab is a unit, and neither a nor b is a zero divisor, prove a and b are units.

Homework Equations

If ab is a unit then (ab)c=1=c(ab) for some c in R.

The Attempt at a Solution

Assume both a and b are not zero divisors, and denote the identity element of R as 1.

Since ab is a unit in R, there exists some c such that (ab)c = 1.
(ab)c=1 \Rightarrow (ca)b = 1 \Rightarrow b is a unit.

Similarly, (ab)c= 1 \Rightarrow a(bc) \Rightarrow a is a unit.

Does the above sufficiently prove the claim in the problem statement? I feel like I'm missing something.

Thanks!

Yes, you should feel uncomfortable with that. You didn't use that a and b weren't zero divisors. If (ab)c=1 then you can regroup that to a(bc)=1. That doesn't prove a is a unit. You also need to show (bc)a=1. Don't assume the ring is commutative.

 

[shamus390]

Yes, you should feel uncomfortable with that. You didn't use that a and b weren't zero divisors. If (ab)c=1 then you can regroup that to a(bc)=1. That doesn't prove a is a unit. You also need to show (bc)a=1. Don't assume the ring is commutative.

Ah thank you, that was a foolish oversight, they wouldn't stipulate a and b weren't zero divisors if it wasn't necessary...

So if we let a, b and c be elements of R such that ab = ac, then:

ab-ac=0_R = a(b-c) = 0_R, and
a is not a zero divisor \Rightarrow (b - c) = 0_R \Rightarrow (b-c) = 0_R \Rightarrow b = c

Does that allow me to claim the following?

Multiplying a(bc) = 1 by a to give a(bc)a = 1a , (1a=a1 by definition) then, a(bc)a = a1 and by cancellation, (bc)a = 1 \Rightarrow a is a unit.

 

[Dick]

Ah thank you, that was a foolish oversight, they wouldn't stipulate a and b weren't zero divisors if it wasn't necessary...

So if we let a, b and c be elements of R such that ab = ac, then:

ab-ac=0_R = a(b-c) = 0_R, and
a is not a zero divisor \Rightarrow (b - c) = 0_R \Rightarrow (b-c) = 0_R \Rightarrow b = c

Does that allow me to claim the following?

Multiplying a(bc) = 1 by a to give a(bc)a = 1a , (1a=a1 by definition) then, a(bc)a = a1 and by cancellation, (bc)a = 1 \Rightarrow a is a unit.

Yes, that's the idea.