Monic Polynomials in $R[X]$

Question

Let $R$ be a ring and consider $f = r_nx^n + 1.x^{n-1} + \cdots + rx + r_0\; \in R[X]$ such that $r^n = 0$ for all $r \in R$. Then can I call $f$ a monic polynomial in $R[X]$ (assume $r_n$ is non-invertible)?

Answer 1

No. A polynomial over $R$ is not the same thing as the function from $R$ to $R$ defined by a polynomial. In particular, if we assume that $r^3=0$ for every $r\in R$ then $$p(x)=2x^3+x^2$$and $$q(x)=x^2$$ define the same function, but (assuming that $2\ne0$ in $R$) they are different polynomials, and in fact $q$ is monic while $p$ is not.

Answer 2

I think the answer is no for the same reason that $x^p-x$ is not said to be the zero polynomial in $\mathbb{F}_p[x]$.

One "reason" that $x^p-x$ is not the zero polynomial in $\mathbb{F}_p[x]$ is that, even though it evaluates to zero in $\mathbb{F}_p$, it does not evaluate to zero in every $\mathbb{F}_p$-algebra—in particular, it's not zero in the tautological evaluation $x\mapsto x$.

Similarly, even though the leading term of your polynomial evaluates to zero in $R$, it doesn't evaluate to zero in every $R$-algebra.