Counterexample for Subring and Units Statement

[erogard]

Hi, I am trying to find a counterexample to disprove this statement, but can't find any:

If S is a subring of a commutative ring R, then $$U(S) = U(R) \cap S$$

Note that U(X) denotes the set of all the units of a ring X, where x is a unit if x has an inverse in X, such that x times its inverse gives 1, the multiplicative identity.

I've tried with integers, quotient groups of integers, complex numbers, etc. but the statement holds for all the cases I've considered.

Any suggestion?

 

[Dick]

Have you tried the rationals, Q? What are the units?

 

[erogard]

Have you tried the rationals, Q? What are the units?

So U(Q) would be all non zero element, including Z without 0.

Then U(Z) = U(Q) intersection Z, which gives Z without 0. But we know that U(Z) = +1 and -1. I think that works!

 

[Dick]

So U(Q) would be all non zero element, including Z without 0.

Then U(Z) = U(Q) intersection Z, which gives Z without 0. But we know that U(Z) = +1 and -1. I think that works!

I KNOW it works.