# Polynomial rings - zero divisors in R and r[x]

• Dec 25th 2012, 03:34 PM
Bernhard
Polynomial rings - zero divisors in R and r[x]
Show that if R has no zero divisors then R[x] also has no zero divisors.
• Dec 25th 2012, 04:11 PM
Bernhard
Re: Polynomial rings - zero divisors in R and r[x]

Let \$\displaystyle f(x) = a_n x^n + ... + a_1 x + a_0\$

and \$\displaystyle g(x) - b_n x^n + ... + b_1 x + b_0 \$

Then \$\displaystyle f(x)g(x) = c_{m+n} x^{m+n) + c_{m+n- 1} x^{m+n-1) + ... c_1 x + c_0\$

where \$\displaystyle c_k = a_kb_0 + a_{k-1}b_1 + ... + a_1b_{k-1} + a_0b_k\$

But \$\displaystyle c_k\$ cannot equal zero since all of the \$\displaystyle a_i b_j \ne 0\$ since R has no zero divisors

Thus R[x] has no zero divisors

Can someone please confirm that this solution is OK?

Peter
• Dec 25th 2012, 07:18 PM
Deveno
Re: Polynomial rings - zero divisors in R and r[x]
well, no.

ck is a sum of elements, so there is no reason to suppose it can't be 0, even if all the terms in the sum are non-zero.

however, you're close...

cm+n = ambn ≠ 0, since am ≠ 0, and bn ≠ 0.

since this is the leading coefficient of f(x)g(x), f(x)g(x) cannot be the 0-polynomial.