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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?

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  • $\begingroup$ Given a $k$-algebra $A$ and a (proper) ideal $I$ of $A$, there is a natural map $k \to A \to A/I$. Now $A/I$ does not really contain $k$, but the image of $k$ under that natural map is an isomorphic copy of $k$. This is the reason that a $k$-algebra is often defined not as a ring containing $k$ as a subring, but as a ring $A$ together with a ring homomorphism $k \to A$. $\endgroup$ Commented Nov 3, 2016 at 8:06

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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.

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