I have been reading Algebra Chapter $0$ by Aluffi and I'm struggling to understand the following:
First, the author proves the lemma:
Let $f(x)$ be a monic polynomial, and assume $$f(x)q_1(x)+r_1(x)=f(x)q_2(x)+r_2(x)$$ with both $r_1(x)$ and $r_2(x)$ polynomials of degree $< \deg f(x)$. Then $q_1(x) = q2(x)$ and $r_1(x) = r_2(x).$
Then it is claimed that this lemma can be summarised as follows:
Assume then that $R$ is a commutative ring. If $f(x)$ is monic then for every $g(x)\in R$ there exists a unique polynomial $r(x)$ of degree $<\deg f(x)$ and such that $$g(x)+(f(x))=r(x)+(f(x))$$ as cosets of principal ideal $(f(x))$ in $R[x]$.
How can I see that the latter statement follows from the lemma?