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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