Consider a commutative ring $R$ with unity. How does one show that $R$ is isomorphic to the quotient of a polynomial ring $\mathbb{Z}[x_1,x_2,\dots]$ (possibly infinitely many variables)?
I have been trying to think of an explicit map, but cannot come up with one. Any help/hint will be appreciated.
Take any element $r\in R$ and consider
$ \mathbb{Z}[r : r\in R]\to R $
That sends each $r\mapsto r$. The map is defined by linearity and it’s a well defined homomorphism and surjective, so that $R$ is a quotient of $\mathbb{Z}[r: r\in R]$