Thread: Integral domain and unit of a ring

Lt R be a commutative ring not neccesarily an integral domain. Determine the truth or falsity of the following statements. In each case, give a proof or find a counter-example

Let x, y belongs to R
(i) x.y in R* ---> x in R* , y in R* (R* = R\{0})
(ii) x.y in U(R) ---> x in U(R), y in U(R)
(iii) x.y in R*\ U(R) ---> x in R*\ U(R), y in R*\U(R)

This is my attempt

(i)If x.y in R* then we have x.y not equal to 0 implies that x is nonzero and y is nonzero. So, the statement is true

(ii)I think this statement is false but I dont know how to prove it

(iii)This part I dont know

thank you

(i)If x.y in R* then we have x.y not equal to 0 implies that x is nonzero and y is nonzero. So, the statement is true

Yes

(ii)I think this statement is false but I dont know how to prove it

Well means there is such that The multiplication in a ring is associative; so what can you deduce from these equalities?

(iii)This part I dont know

Try to see what happens if you take

(ii)I think this statement is false but I dont know how to prove it

what can you deduce from these equalities?

It implies that xy is the inverse of a, and since xy in U(R) then x in U(R) and y in U(R). Hence the statement is true

Originally Posted by knguyen2005

(ii)I think this statement is false but I dont know how to prove it

what can you deduce from these equalities?

It implies that xy is the inverse of a, and since xy in U(R) then x in U(R) and y in U(R). Hence the statement is true

I don't think your argument is valid here. I think you're supposed to deduce that and . You can do this since is a commutative ring. From those equalities, you get

Originally Posted by clic-clac

Try to see what happens if you take

You can't take , as this is a unit. We are looking at the non-zero ring element which are not units.

What we want to prove is the assertion is wrong, i.e. can be non invertible with or invertible.

With it becomes clear that whenever there are non zero non invertible elements in then the implication is false.


Edit: Quite close indeed

Originally Posted by clic-clac

What we want to prove is the assertion is wrong, i.e. can be non invertible with or invertible.

With it becomes clear that whenever there are non zero non invertible elements in then the implication is false.

I know - I'm having a slow evening, and I couldn't get to my laptop in time to change it!

EDIT: Although, by the timing of this post it looks like I was pretty close!