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the leading term of a module

I'm reading CLO and I have a question about the following Prop: Let I be an ideal in a polynomial ring $k[x_1,\ldots,x_n]$. Then $k[x_1,\ldots,x_n]/I$ is isomorphic as a $k$-vector space to $S = ...
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How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...