I have trouble with understanding this sentence in the context of polynomial rings: "We will assume that polynomials satisfy right evaluation. That is, a polynomial can only be evaluated once it is written so that powers of $x$ appear to the right of their coefficients".
I don't underatand this statement. I really appreciate if someone could explain it to me. Thanks in advance.
(clarification from comments: this is from "Null ideals of subsets of matrix rings over fields" by N.J.Werner, and the polynomials are over a noncommutative ring)
They define the evaluation map $ev_r$: $\ldots \to R$ only on polynomials of form $f(x) = \sum\limits_{i=0}^n a_i x^i$ with $f \mapsto \sum\limits_{i=0}^n a_i r^i$. Unlike the commutative case, this definition need not extend to a homomorphism $R[x] \to R$, hence the need for a convention.
Edit: Noncommutative rings and the evaluation homomorphism