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Your favorite way to think of $k[x_1,ldots,x_n]$ modulo some graded ideal?

I was looking at Graded Syzygies by Peeva, and on page 3 she says "throughout this book, $R=S/I$ and is standard graded." Here, $S=k[x_1,ldots,x_n]$ and $I$ is some graded ideal in $S$. I understand that this simply yields a generalization of free $S$-modules, since we could just take $R=S/(0)cong S$. But what is the most intuitive way to think about the ring $S/I$? Im curious how researchers in the fields of algebraic geometry and commutative algebra intuit this ring. For example, I know that most people think of $k[x]/(x^2)$ as killing off any terms with degree greater than or equal to $2$, but is there a nice way this idea carries over to $S/I$? I know this question is slightly subjective, but I feel that some input would be useful to other users of the site.

Mathematics Asked on November 12, 2021

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