# Tagged Questions

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### The spectrum of a commutative ring with unity and its “topology”

Let $\operatorname{Spec}(R)$ be the set of prime ideals in the commutative ring with unity $R$, and let $\mathfrak a$ be some ideal. Show that we get a topological space if we define the closed sets ...
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### Subbase of a topology containing prime ideals (commutative ring)

Let $A$ be a commutative ring. Prove that the set of the ideal primes of $A$, along with $A$, is a subbase of some topology on (the subjacent set of) $A$ and that the complements of the prime ideals ...
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Let X be a finite partially ordered set. How can to prove that there exists a ring R such that Spec R ≅ X? If anyone has any good way of thinking about them do please divulge..
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### The Zariski topology on $\operatorname{Spec} A$ as an intial topology

Given any commutative ring $A$ let $\operatorname{Spec} A$ be the space of prime ideals of $A$. Can we interpret the Zariski topology as an initial (or final) topology with respect to some canonical ...
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### Irreducible components of $Spec(A)$

A topological space $X$ is called irreducible if given $A_{1}, A_{2}$ open sets $\neq \emptyset$ then $A_{1} \cap A_{2} \neq \emptyset$. The maximal irreducible topological subspaces of $X$ are ...
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### Is there a topological characterisation of non-Archimedean local fields?

A local field is a locally compact field with a non-discrete topology. They classify as: Archimedean, Char=0 : The Real line, or the Complex plane Non-Archimedean, Char=0: Finite extensions of the ...
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### Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
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### A property of $I$-adic topologies

Let $R$ be a commutative ring with multiplicative identity and $I$ a proper ideal of $R$ such that the intersection of its powers is the zero ideal. It can be shown that if the $I$-adic topology is ...
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### Noetherianty of ring of continuous functions [duplicate]

Let $X$ be a topological space. Is there any topological property on $X$ that be equivalent to $C(X,\mathbb R)$ be noetherian ring?
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### When is the ring of continuous functions Noetherian?

Let $X$ be a topological space. Is there any topological property on $X$ that be equivalent to $C(X,\mathbb R)$ being a noetherian ring?
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### Is there a simple way to state continuity for $I$-adic topology?

Let $R$ be a commutative ring with the $I$-adic topology defined by an ideal $I$, and let $S$ be a commutative ring with the $J$-adic topology for an ideal $J$. How would you translate saying that a ...
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### Relation between spectrum of a ring and its quotient ring and localization.

Let $A$ be a commutative ring. $I$ be an ideal of $A$, $S$ be a multiplicative closed subset. We know that : there is 1-1 correspondence between the prime ideals $\mathfrak{p}\in Spec A$ containing ...
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### Is there an open mapping theorem for affine morphisms?

Let $A$ and $B$ be rings. If $\varphi : A \longrightarrow B$ is such that $^a\varphi : Spec(B) \longrightarrow Spec(A)$ is bijective, then in what conditions $^a\varphi$ is a homeomorphism? Or, more ...
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### studying the topology of a real algebraic set

Let $f_1,\ldots,f_n \in \mathbb{R}[x_1,\ldots,x_m]$ be polynomials with real coefficients and let $I$ be the ideal that they generate. Denote by $V_{\mathbb{R}}(I)$ the corresponding real variety, ...
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### $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus \{0\}$ are not homeomorphic

Let $k$ be an algebraic closed field. Why $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus\{0\}$ (for $n>1$) are not homeomorphic with respect to the Zariski topology?
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### Looking for a “prime-ish” family of subsets

Is there a nontrivial (what I mean is below) example of a compact Hausdorff space $X$ and a family $\mathscr{F}$ of subsets of $X$ with the following pair of properties? $\mathscr{F}$ is ...
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### limits of sequences of topological rings

Let $A$ be a ring and $I$ an ideal of $A$ such that $A$ is complete in the $I$-adic topology. Let $a \in I$. Then the sequence $y_n=1-a+a^2-a^3+\cdots+(-1)^n a^n$ converges in $A$. By definition of ...
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### Showing that the natural map into the completion is continuous

Let $M$ be an $A$-module and $M=M_0 \supset M_1 \supset \cdots$ a sequence of submodules, which we define to be a fundamental system of neighborhoods of $0$. Thus we make $M$ into a topological group. ...
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### algebraic distance of an element of a ring from an ideal

Let $A$ be a commutative ring and $I$ an ideal. Does there exist a notion of "distance" of an element $x \in A$ from the ideal $I$? This "distance", need not be of the form $A\rightarrow \mathbb{R}$; ...
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### Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
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### Equivalent definitions of Noetherian topological space

It is well known that we have many different definitions of noetherianity for rings. Namely, given a ring $R$, the following are equivalent: 1) every ideal of $R$ is finitely generated. 2) $R$ ...
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### Completion as a functor between topological rings

In the following all rings are assumed to be commutative and unitary. Preliminaries: For any topological ring $R$ we can form its completion $\widehat{R}$ by taking all Cauchy sequences modulo null ...
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### What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?

Let $R$ be a commutative ring with a unit. $\newcommand{\spec}{\operatorname{Spec}}\spec(R)$ denotes the set of all prime ideals in $R$, and it can be topologized using the Zariski topology. Last ...
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### Irreducibility and Subspace Topology

Is the following statement true? "Let $X$ be a topological space, $Y$ a subspace and $S$ a closed and irreducible subset of $X$. Then $Y \cap S$ is not necessarily irreducible in $Y$." Counterexamples ...
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### Combinatorial Dimension of a Topological Space - Ascending versus Descending Chains

The (combinatorial) dimension of a topological space is defined as the supremum of the lengths over all strictly ascending chains of closed irreducible subsets (e.g. Hartshorne). Can it also be ...
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### $\operatorname{Spec}A$ and $T_1$ separation axiom

Let $A$ be a commutative ring. Why $\operatorname{Spec}A$ almost never satisfies the $T1$-separation axiom (Matsumura, Commutative Ring Theory, p.25)?
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### A theorem due to Gelfand and Kolmogorov

For any topological space $X$, we can define $C(X)$ to be the commutative ring of continuous functions $f\,:\,X\rightarrow \mathbb{R}$ under pointwise addition and multiplication. Then $C(-)$ becomes ...
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### What do I need to know to understand the completion of the field of rational functions of a non-singular projective curve?

So the title gives the jist of my question. Specifically, let $X$ be a non-singular projective curve, $P$ a point on $X$, $v_P$ the discrete valuation associated to the ring $\mathcal{O}_P$. Then I ...
### Zariski topology on prime $\mathrm{Spec}$ of a ring $R$
Let $R$ be a commutative unital ring. Let $\mathrm{Spec}(R) = \{ \mathfrak p \subset R \mid \mathfrak p \text{ a prime ideal of } R \}$. We define a set $C$ to be closed in this space if and only if ...