Questions tagged [local-rings]
In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.
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When the derived tensor product of a perfect complex with a degreewise homologically finite bounded below complex is homologically bounded
Let $R$ be a commutative Noetherian local ring. Let $G$ be a perfect $R$-complex which is not acyclic. Let $M$ be a homologically bounded below
complex (i.e., $H_n(M)=0$ for all $n \ll 0$) of $R$-...
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How many finite commutative rings with $(1\neq 0)$ are there for given $|U(R)|$?
I was reading about Unit graphs (Ashrafi, 2010). There are some results based on $|U(R)|$. So I was wondering if for given $|U(R)|$ there are only finite number of rings or as we increase $|R|$ does $|...
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Is "being a local ring" a local property?
Setting and Definition
Let $A$ be a commutative unital ring. If we say $A$ is a local ring, we mean that $A$ has only one maximal ideal.
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Is it true that
\begin{align}
A\text{ is a local ring ...
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On ideal $I$ such that tensoring the free cover of maximal ideal with $R/I$ is an isomorphism
Let $I$ be a nonzero proper ideal of a Noetherian local domain $(R, \mathfrak m)$ of dimension $1$. Assume $I$ is not a principal ideal. Let $n$ be the minimal number of generators of $\mathfrak m$, i....
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On a particular type of specialization-closed subset of prime spectrum of reduced local ring of Krull dimension $2$
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of Krull dimension $2$. Let $S$ be a specialization closed subset of $\text{Spec} (R)$ such that $P \notin S$ for every minimal prime ideal $P$...
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Cohen-Macaulay local ring modulo its nilradical
Let $(R, \mathfrak m)$ be a local Cohen-Macaulay ring with nilradical $N$. Must $R/N$ also be Cohen-Macaulay?
The answer is obviously yes when $\dim R=0$. The answer is also yes when $\dim R=1$ as ...
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Henselian ring as filtered colimit
I would like to apply Artin's Approximation Theorem. The setup is the following. Let $A$ be local Henselian ring. Then why is the following claim correct: $A$ is the filtered colimit
$$
A=\varinjlim ...
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Ring structure of $\mathbb{F}_q[x]/(f)$
Can we classify rings $\mathbb{F}_q[x]/(f)$ for $f\in\mathbb{F}_q[x]$?
By the Chinese remainder we have the decomposition
$$
\mathbb{F}_q[x]/(f) \cong \mathbb{F}_q[x]/(f_1^{e_1}) \oplus \mathbb{F}_q[x]...
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Jacobson radical of associative algebra, module finite over its center which contains a local subring [closed]
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring and let $S$ be a module finite associative $R$-algebra such that $R$ is a subring of the center of $S$. Then, is it true that $\mathfrak mS$...
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Non-free projective modules over polynomial rings with local coefficients?
(Tell me if this question has been phrased appropriately as I'm first time asking question here)
Here we only consider commutative unital rings.
Q: Are there examples of non-free (say f.g.) projective ...
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On extending minimal set of generators of an ideal to a bigger ideal
Let $J\subseteq I$ be ideals of a commutative Noetherian local ring $(R,\mathfrak m)$ such that $J\cap I\mathfrak m=J\mathfrak m$. This condition implies that every minimal generating set of $J$ is ...
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Let $R$ be an integral domain, $\mathfrak{p}$ a maximal ideal with finite index. Are the ideals in $R/\mathfrak{p}^r$ totally ordered?
Let $R$ be an integral domain, $\mathfrak{p}$ a prime ideal with finite index (since we are in an integral domain, this is equivalent to assuming $\mathfrak{p}$ is a maximal ideal with finite index). ...
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Estimating Length/Multiplicity of an Artinian Ring
I have a commutative graded Noetherian ring of the form $$S=k[x_0,…x_r, a_0,…,a_r]/(I_1+I_2+J)$$ which I am localizing at $(x_0,…,x_r)$; in my case, this is a minimal prime, so the result is an ...
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Algebraic Hartogs's lemma for dimension 1 rings
I am reading the proof of what Vakil calls "algebraic Hartogs's lemma" which says that if $A$ is an integrally closed, integral, Noetherian ring, then $$A=\bigcap A_{\mathfrak{p}},$$ where ...
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A good reference for the graded version of Nakayama's Lemma
I am looking for a reference for a graded version of Nakayama's Lemma. Specifically, if $A$ is a local graded-commutative ring with unique maximal ideal $I$, $f:M\to N$ is a morphism of finitely ...
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