# Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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### The relation between been the quotient ring of a prime ideal and its localization

Let $A$ be a ring and $\mathfrak{p} \subset R$ be a prime ideal. Set $A_\mathfrak{p}=R[U^{-1}]$, where $U= A-\mathfrak{p}$. What is the relation between $A/\mathfrak{p}$ and $A_\mathfrak{p}$? My ...
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### Tensor product of domains is a domain

I'm reading Milne's Algebraic Geometry course notes, version 5.22, as a companion to an algebraic geometry course I'm taking now. Proposition 4.15 states: Let $A$ and $B$ be $k$-algebras, which are ...
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### Vanishing Ideal of a Linear Subspace

Let $F$ be an infinite field. Let $V$ be a subspace of $F^n$. Let $V^{\perp}$ be the set of all linear functionals $F^n \rightarrow F$ that vanish on $V$. Let $I(V)$ be the vanishing ideal of $V$, i.e....
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### Why is the localization at a prime ideal a local ring?

I would like to know, why $\mathfrak{p} A_{\mathfrak{p}}$ is the maximal ideal of the local ring $A_{\mathfrak{p}}$, where $\mathfrak{p}$ is a prime ideal of $A$ and $A_{\mathfrak{p}}$ is ...
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### Is every Artinian module over an Artinian ring finitely generated?

I know that if $R$ is Artinian, then a f.g. $R$-module is Artinian. Is f.g. a necessary condition?
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### Motivation behind the definition of Zariski tangent space

Intuitively I think of tangent space at a point as the set of all points lying in the tangent plane passing throug that point. Here is the definition of Zariski tangent space Let X be an ...
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### Question on an isomorphism in the proof that $k[V \times W] \cong k[V] \otimes_k k[W]$

First I should say that I am aware of the existence of this question here and this question here. My question is a little different from these two because I am asking about a certain detail in the ...
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### What does projective space classify?

Let $A$ be a ring and let $\mathbb{P}^n = \operatorname{Proj} \mathbb{Z} [x_0, \ldots, x_n]$. Question. What does $\mathbb{P}^n$ classify? In other words, is there some kind of algebraic structure (...
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### Are finitely generated projective modules free over the total ring of fractions?

Let $Q(A)$ be the total ring of fractions of a commutative reduced non-noetherian ring $A$. Let $P$ be a finitely generated projective module over $Q(A)$ which is of constant rank (i.e. locally free ...
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### Lorenzen embedding theorem for an $\ell$-group

The Lorenzen embedding theorem for an lattice-ordered group says that any lattice-ordered group can be embedded into a product of totally ordered group. What condition on lattice-ordered group makes ...
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### Intersection of finitely generated ideals

Let $I$, $J$ be finitely generated ideals in a ring $A$ (commutative with identity). I know that the intersection need not be finitely generated: can somebody give me an example? Thanks.
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### Minimal syzygies for polynomial ideals

Let $I$ be an ideal of $S=k[x_1,\dots,x_n]$. I am asked to find a minimal free graded resolution of $I$, by means of syzygy matrices. I suppose there has to be an algoritmic approach to it, provided ...
Let $u = (u_1 , \ldots , u_n ) \in \mathbb{A}^n$. Let $I$ be the ideal of $A = \mathbb{C}[x_1 , \ldots x_n ]$ generated by the elements $x_1 - u_1 , \ldots , x_n - u_n$. (i) Show that as a $\mathbb{C}... 1answer 258 views ### vector bundles on the affine line over a PID Let$R$be a PID. Is every finitely generated projective$R[T]$-module free? In other words, is every vector bundle on$\mathbb{A}^1_R$trivial? For$R=k[X]$this is true by the Theorem of Quillen-... 1answer 486 views ### Coordinate ring of a cartesian product I am considering the coordinate ring$k[X \times\mathbb{A}^n]$, where$X$is an algebraic variety in$\mathbb{A}^n$. I want an isomorphism between this and the polynomial ring$k[X][y_1,\ldots, y_n]$. ... 1answer 105 views ### Noetherian modules Question: Let$R$be a Noetherian ring, and$M$be an$R-$module, show that$M$is Noetherian if and only if$M$is finitely generated. This is a question on my homework, I'm really confused about ... 1answer 68 views ### practical condition for minimality in primary decomposition Situation:$I$is an ideal in a polynomial ring with a primary decomposition, not necessarily minimal (minimal=irredundant). I want to minimal-ize it. For any couple of primary ideals with the same ... 2answers 293 views ### Dimension of the total ring of fractions of a reduced ring. Let$A$be a commutative reduced ring (need not be noetherian). Let$S$be the set of all non-zerodivisors of$A$. What is the Krull dimension of$S^{-1}A$? 3answers 722 views ### Noetherian Local Ring I came across this old exam problem. If$R$is a local Noetherian ring and$I$is an ideal in$R$such that$I^2=I$then$I =0$. Any hint would be appreciated. I'm only familiar with what the ... 1answer 208 views ### Hypotheses of the Conormal Exact Sequence On Wikipedia, in the description of the conormal exact sequence, it is described as arising from a closed immersion, which corresponds in the affine case to a surjection of algebras. However, in ... 3answers 635 views ### The vanishing ideal$I_{K[x,y]}(A\!\times\!B)$is generated by$I_{K[x]}(A) \cup I_{K[y]}(B)$? Let$K$be a field,$x=(x_1,\ldots,x_m)$,$y=(y_1,\ldots,y_n)$,$A\!\subseteq\!\mathbb{A}^m_K$,$B\!\subseteq\!\mathbb{A}^n_K$. Does there hold $$I_{K[x,y]}(A\!\times\!B)=\langle\langle I_{K[x]}(A) \... 3answers 658 views ### Localization of Noetherian and Artinian Modules Theorem: Let R be a commutative ring with unity, and S\subset R be a multiplicatively closed subset. If M is a Noetherian (Artinian) R-module then S^{-1}M is Noetherian (Artinian) S^{-1}R-... 2answers 593 views ### Companion Lecture Notes to Atiyah-MacDonald? Is there a set of lecture notes that follow Atiyah-MacDonald and expand on the dense passages, point out typos and so forth? 1answer 185 views ### Prime ideal in a Dedekind domain Let \mathfrak p be a prime ideal in a Dedekind domain O with field of fractions K. Define$$\mathfrak p^{-1}= \{x \in K: x\mathfrak p \subset O\}.$$Let \mathfrak a \subset \mathfrak p and \... 2answers 111 views ### Zerodivisors and nilpotents in A/I I'm studying primary decomposition in the case of polynomial rings with coefficients in a field. I have defined associate prime ideals of an ideal I as the radicals of the primary ideals appearing in ... 1answer 674 views ### Finite + surjective + projective implies flat? Let f: X \rightarrow Y be a morphism of irreducible projective varieties, that is both finite and surjective. Does this mean that it is flat? I have tried the following: By finiteness, the map is ... 1answer 119 views ### Generators for this ideal I have a ring R unitary and commutative with four elements and characteristic 2. I have$$I=\{f \in R[X,Y]; f(t,t^2)=0\ \forall t \in R\}.$$I have to find a finite number of generators for this ... 1answer 180 views ### A valuation ring In Qing Liu, Algebraic Geometry and Arithmetic Curves, page 116, exemple 4.1.8, one has$\mathcal{O}_K$a discrete valuation ring with uniformizing parameter$t$,$P\in\mathcal{O}_K[S]$an Eisenstein ... 4answers 3k views ### A proof that shows surjective homomorphic image of prime ideal is prime Let$A, B$be commutative rings with$1_{A}, 1_{B}$. Suppose that$\mathfrak{p} \neq (1)$is a prime ideal in$A$with$\mathfrak{p} \supseteq \ker{\varphi}$where$\varphi: A \rightarrow B$is a ... 4answers 154 views ### Irreducible element of the ring. Element$X_1 X_2 \cdots X_n - 1$is irreducible in$K[X_{1},\ldots,X_{n}]$for$n\ge 1$, where$K$is a field. For$n=2,3$it is easy to see that the element is irreducible but for higher value of$n$... 2answers 462 views ### Elements of the square of a prime ideal Let$R$be a commutative ring with unity, and let$\mathfrak{p} \subset R$be a prime ideal. If$ab \in \mathfrak{p}^2$, does one of the following hold?$a \in \mathfrak{p}^2$;$b \in \mathfrak{p}^...
Let $K/F$ be a field extension. I am interested in the situation where there exists a field extension $L/F$ such that the ring $L \otimes_FK$ is not a field. If there exists $z\in K \setminus F$ ...