> Hi,

> I am confused by problem 10, section 3.2 from Herstein.
> The stated problem is:
> "Show that the commutative ring D is an integral domain iff for all
> a,b,c in D and a ne 0, ab=ac implies b=c"

> I read the logic of this proof as follows:

> Let D be a commutative ring  and ( D is an integral domain iff for all
> a,b,c in D and a ne 0, ab=ac implies b=c)

> I am having trouble with the proof of the back implication of this
> proposition.
> In class, we "proved" this proposition as follows:
> Suppose D is a commutative ring.
> Suppose also that for all a,b,c in D, ab = ac implies b=c
> Now, let x and y be elements in D such that x ne 0 and xy = 0
> using the basic ring property that the zero element times any element
> in D is zero, we can say:
> xy = 0 = x0
> which by our hypothesis implies y=0.
> So we have proved that xy=0 and x ne 0 implies y = 0, which implies D
> has no zero divisors (since it is logically equivalent to xy ne 0 or x
> = 0 or y = 0, which implies the statement "xy=0 and x ne 0 and y ne 0"
> is always false and can therefore never occur, hence we can have no
> zero divisors).

> PROBLEM: My problem rests with the fact that (in the proof of the back
> implication) we have showed that D is a commutative ring (given by
> hypothesis) and that D has no zero divisors, but we have not shown
> that D has a unit element, which is the third condition required in
> the definition of "integral domain"

> So I don't see how, in problem 10 from Herstein, we can prove that the
> ring also has a unit element when we are proving the back implication.

> Am I missing something or is Herstein a bit off on this one?