# Extensions of primes, flatness, and intersection flatness

@article{Hochster2020ExtensionsOP, title={Extensions of primes, flatness, and intersection flatness}, author={M. Hochster and Jack Jeffries}, journal={arXiv: Commutative Algebra}, year={2020} }

We study when $R \to S$ has the property that prime ideals of $R$ extend to prime ideals or the unit ideal of $S$, and the situation where this property continues to hold after adjoining the same indeterminates to both rings. We prove that if $R$ is reduced, every maximal ideal of $R$ contains only finitely many minimal primes of $R$, and prime ideals of $R[X_1,\dots,X_n]$ extend to prime ideals of $S[X_1,\dots,X_n]$ for all $n$, then $S$ is flat over $R$. We give a counterexample to flatness… Expand

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