Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extensions of primes, flatness, and intersection flatness   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 over a reduced quasilocal ring $R$ with
infinitely many minimal primes by constructing a non-flat $R$-module $M$ such
that $M = PM$ for every minimal prime $P$ of $R$. We study the notion of
intersection flatness and use it to prove that in certain graded cases it
suffices to examine just one closed fiber to prove the stable prime extension
property.

30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org