Is an Non-Zero Coefficient in a Polynomial a Zero Divisor in R[x]?

[chuy52506]

Let R be a commutative ring. If an doesn't equal 0 and
a0+a1x+a2x^2+...+anx^n is a zero divisor in R[x], prove that an is a zero divisor in R.


What I did was say if the polynomial is a zero divisor in R[x] then let that polynomial equal p(x) and any other polynomial be q(x) with coefficients b0,b1,...,bm, then p(x)*q(x)=0. And the leading coefficient and degree will be an*bm*x^(n+m) which will be a zero divisor in R. Therefore an will be a zero divisor. However I don't know what to say to show this? is it correct?

 

[micromass]

What you said sounds good! What more do you want to show??

 

[chuy52506]

Haha I'm not sure if it's legit?

 

[micromass]

Yes,, it's legit, but you'd have to watch your notations!