Show that if R has no zero divisors, then R[x] has no zero divisors.
Since R has no zero divisors, then such that , .
Is this next part correct (not sure what to do next)?
@HallsofIvy:
You're absolutely right, but is it not enough to say that the coefficients of the polynomial ring R[X] are ?
And I think dwsmith means with:
That's how I interpreted it.
The first thing that comes to my mind is to prove the contrapositive statement. That is I'm going to prove that if has any zero divisors the so does . Here goes:
Let such that and (the zero polynomial). We will write
and .
Now since and are nonzero then there exist some coefficients and such that (these coefficients are in ). Now we multiply and to get
.
But we know that the only way for to be the zero polynomial is if all its coefficients are , and since all the coefficients are zero then , but so and must be zero divisors. qed
A little simpler than obd2 but it's the same idea :
such that
Let and (those exist because of the non nullity of and ). Let now .
One have .
So is the coefficient of . And so it is null : . So is or .
If , it's a contradiction with the definition of . If , it's a contradiction with the definition of .
QED.
P.S. : I'm new on this forum, and I love the integration. It's really more efficient and beautiful than in other forums.