Q:

Why the quotient ring of a proper polynomial ideal is a k-algebra？

I'm reading Cox, Little and O'Shea's Ideals, Varieties, and Algorithms on my own.
In page 277, the authors define k-algebra as follows: A k-algebra is a ring which contains the field k as a subring. In page 278, the authors say "Examples of k-algebras include ... the quotient ring of a proper polynomial ideal."
My question is, since elements of a quotient ring are sets, why the quotient ring of a proper polynomial ideal can contain the field k as a subring?

A:

Containing a field $k$ means nothing else but having a (unital) ring homomorphism from $k$ into the ring.
In your case, we obtain the homomorphism as the following composition:
$$k \hookrightarrow k[x_1, \dotsc, x_n] \to k[x_1, \dotsc, x_n]/I$$
, where the second map is the canonical projection.

