All Questions
Tagged with local-rings ideals
7 questions
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Symbolic power of an ideal associated to non-singular algebraic set
Let $Z\subset \mathbb P^n$ be a reduced non-singular algebraic set and $I$ denote the saturated homogeneous ideal of $Z$. I have seen the following result without proof:
For all $ n\geq 1$, $I^{(n)}=(...
- 1,713
1
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1
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465
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Valuation ring whose maximal ideal and every ideal of finite height are principal
Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
user111492
1
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2
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194
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The quotient of an algebra with an ideal whose generators are decomposed as the product of irreducible elements
I would like to find reference for the following statement.
I need it only in the particular case when $A=\mathcal{O}_{(\mathbb{C}^n, 0)}$ is the local algebra of holomorphic germs $(\mathbb{C}^n, 0) \...
- 73
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When is $rad(L)[x_1,\ldots]$ radical in $Ker(\varphi_\ast)$?
Suppose we have a local ring $L$ (not necessarily commutative) such that $L/rad(L)$ is a division algebra (here $rad(L)$ is the Jacobson radical of $L$). We clearly have the canonical surjection $\...
- 98
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85
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Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?
Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
- 480
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96
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Length of generic intersection in local ring
Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.
If $a$ that is not a zero divisor of $R/I$ we have ...
-2
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2
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764
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Reduced ring with all non-prime ideals finitely generated
Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ?
Without reduced assumption, it is not true even ...
user111492