论文标题
素数，平坦度和交点平坦的扩展
Extensions of primes, flatness, and intersection flatness
论文作者
论文摘要
我们研究$ r \ to s $具有$ r $的主要理想的属性，即$ r $的理想或$ s $的单位理想，以及该物业与两个戒指相同的不确定之后，该物业继续存在的情况。我们证明，如果减少$ r $，$ r $的每个最大理想仅包含$ r $的最低数量，而$ r [x_1，\ dots，x_n] $的主要理想是$ s [x_1，\ dots，\ dots，x_n] $ for $ n $ for $ n $，则$ n $ s $ s $ r $ r $ r $。我们通过构建一个非flat $ r $ r $ -Module $ m $ $ $ r $的较小的准环$ r $对平面度进行了反例，从而使每个最小的prime prime prime $ p $ $ r $ $ $ m = pm $。我们研究了交叉平坦度的概念，并使用它来证明，在某些分级情况下，仅检查一个封闭的纤维以证明稳定的Prime扩展特性就足够了。
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.
