Dimension of $R\left/aR\right.$.
In my algebra course I was asked to solve the following problem:Let $R$ a finite type $K$-algebra and suppose $R$ is an integral domain. If $0\neq a\in R$ is not invertible show that $\dim...
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I am trying to prove this statement by deducing a contradiction to Krull's Principal ideal theorem. Here is my attempt:Denote $R$ and $P$ the ring and the prime ideal under consideration, respectively,...
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Let $A$ an integrally closed domain and $B$ a commutative ring extension of $A$ that is finitely generated as an $A$-module.For $f\in B$ is it true that there exists $a_f\in A$ s.t. $\sqrt{(f)\cap...
View ArticleQuestions about the proof of Theorem 13.4 in Matsumura's Commutative Ring Theory
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View ArticleIs $\dim (M/xM) = \dim M - 1$ for some $x \in m$ implies $x$ is an...
Let $R$ be a Noetherian local ring with maximal ideal $m$, and let $M$ be a finitely generated $R$-module. It is known that if $x \in m$ is an $M$-regular element, then the dimension of $M/xM$ is equal...
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