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Let $k$ be a field and let $A \neq 0$ be a finitely generated $k$-algebra, and $x_1, \cdots, x_n$ generate $A$ as a $k$-algebra. Is there any relationship(inclusion, homomorphism, etc.) between $A$ and $k[x_1,\cdots,x_n]$?

How about the case if $x_1,\cdots,x_n$ are algebraically independent over $k$?

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If $x_1, \dots, x_n$ are elements of $A$, $k[x_1, \dots, x_n]$ usually indicates the subalgebra of $A$ generated by $x_1, \dots, x_n$ (in your case $A$).

Let us write $k[X_1, \dots, X_n]$ for the polynomial algebra on the variables $X_1, \dots, X_n$. Then $A$ is isomorph to a quotient of $k[X_1, \dots, X_n]$, more precisely one can consider the homomorphism $$ \begin{array}{c} \varphi: k[X_1, \dots, X_n] \to A\\ p(X_1, \dots, X_n) \mapsto p(x_1, \dots, x_n) \end{array} $$

$\varphi$ is surjective by hypothesis and you have $A \cong k[X_1, \dots, X_n]/\mbox{Ker}\,\varphi$.

Algebraic independence is equivalent to the condition $\mbox{Ker}\,\varphi = (0)$. So, in this case, $A$ is isomorph to $k[X_1, \dots, X_n]$.

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