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

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Showing that an element generates the kernel

$I $ is a monomial ideal generated by $\left < m_1, \dots, m_n\right >$ and suppose we also have an $R$-module homomorphism $\phi: \oplus_{j = 1}^n Re_j \to I$ defined by $$\phi(e_i) = m_i.$$ ...
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36 views

Is a divisible module over a local principal ideal domain a torsion module?

Is an injective module over a local principal ideal domain a torsion module? We know that injective modules and divisible modules over a PID are equivalent. What do we say about the torsion submodule ...
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48 views

System of parameters in a Noetherian local ring

I'am studying "system of parameters" on Commutative Ring Theory by H. Matsumura. There is a theorem about height of an ideal and the number of generators of the ideal. In the proof of the part (ii), ...
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29 views

showing $\bar{X}$ is irreducible in $\mathbb{R}[X,Y]/(X^2+Y^2+1)$ [duplicate]

Consider the ring $A=\mathbb{R}[X,Y]/(X^2+Y^2-1)$. Let $\bar{X}$ be the image of $X$ in $A$. Show that $\bar{X}$ is irreducible in $A$. I tried this by assuming that ...
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54 views

Are the following monomial modules?

If $I$ is an monomial ideal of $R$ and $M, N$ are monomial modules of $\oplus_{i = 1}^{r} Re_i$, then the following are monomial modules. Why? 1) $M + N = \{m + n: m \in M, n \in N \}$ (because it's ...
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The ring of polynomials in $X, Y$ all of whose partial derivatives with respect to $X$ vanish for $Y=0$ is Noetherian? [duplicate]

The ring of polynomials in $X,Y$ all of whose partial derivatives with respect to $X$ vanish for $Y=0$ is Noetherian ?
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37 views

When localization preserve depth?

If I have (I,M), if we localize I with some prime ideal in the support of M, then the depth (the module becomes $M_p$) will be larger or equal to the original depth of I. My question is when the ...
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Number of generators of ideal if quotient field has certain transcendence degree

I am trying to prove the following statement by induction on $n$: Let $P$ be a prime ideal of $\mathbb{Z}[X_1,\ldots,X_n]$ with $\mathbb{Z}\cap P = \{0\}$. Suppose that ...
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40 views

quasi-coherent sheaf on an affine scheme and extension of scalars

Suppose $\widetilde{M}$ is an quasi-coherent sheaf on $SpecA$. $SpecB$ is an open affine subscheme of $SpecA$. Is it true that $\widetilde{M}(SpecB) \cong M \otimes_A B$?
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18 views

System of rational polynomial equations with complex root also has a solution of algebraic numbers [duplicate]

Consider a system of equations $$f_1(x_1,...,x_k)=0,...,f_n(x_1,...,x_k)=0$$ where $f_1,...,f_n$ are polynomials in $\mathbb{Q}[x_1,...,x_k]$. Suppose the system has a solution in $\mathbb{C}^k$. ...
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Every non-Noetherian module has a submodule maximal with respect to being not finitely generated. [duplicate]

Let $M$ be a module. Show that if $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated whenever $N<A\leq M$. The question is related to If $M$ isn't ...
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29 views

Field of definition of an Ideal

I am trying to prove the following statement from Introduction to Commutative Algebra and Algebraic Geometry by Ernst Kunz, p.16, Q9. Let $I$ be an ideal of the polynomial ring $K[X_1,\dots,X_n]$ ...
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21 views

Does Magma let you specify primary invariants?

I am cross-posting this question from scicomp.SE. The computer algebra system Magma can calculate primary invariants (i.e. a homogeneous system of parameters) in an invariant ring of a finite group ...
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110 views

Why is the map from $A^n$ to $M$ a surjective homomorphism?

How can one do the problem 1.3.11 b in Algebraic Geometry and Arithmetic Curves? I have read basics of commutative algebra but this one seems to be too difficult. Let $A$ be a commutative ring with ...
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62 views

Groebner basis and prime ideals.

Let $I$ be an ideal in a polynomial ring $P = K[x,y_1,\dots,y_n]$ and assume that $I \cap K[x]\neq (0)$. Let $>$ be an elimination ordering for $\{y_1, \dots, y_n\}$ and $G$ is a Groebner basis for ...
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66 views

Spectrum and maximal spectrum of a ring

How do the $\mathrm{Spec}(\mathbb{C}\left [ X \right ])$ and $\text{m-Spec}(\mathbb{C}\left [ X \right ])$ look like? I understand the definitions of $\mathrm{Spec}(R)$ and $\text{m-Spec}(R)$ for a ...
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23 views

Relation between minimal primes of a Noetherian graded ring and its subring

Let $A=⊕A_i$ be a Noetherian graded ring. Is there any relation between minimal primes of $A$ and minimal primes of $A_0$ (its $0$-th component)? In fact, my motivation is tight closure theory. I ...
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35 views

Poincaré series pole at $1$

Let $A$ be a graded ring and $M$ a graded $A$-module. By $P(M,t)$ we denote the Poincaré series for $M$. In Atiyah and Macdonald, theorem 11.1 claims $P(M,t)=\dfrac{f(t)}{\prod _{i=1}^n ...
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37 views

Does every non-archimedean absolute value on field take value in $\mathbb{Q}$

Let $K$ be a field, a non-archimedean absolute value is defined to be a map $K\to \mathbb{R}$ satisfying $|x|=0\Rightarrow x=0$, $|x|\cdot|y|=|xy|$ and $|x+y|\leq\max(|x|,|y|)$. Is there an example ...
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Origin of the name “Cauchy sequence” in abstract algebra

I know the origin in analysis as it is derived from our good old chap Cauchy himself. However I have read about Cauchy sequences in algebra, I'll use groups for this one, let $G$ be a group and ...
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89 views

best books, lecture notes, for studying pullback rings

Does anyone have suggestions for books, or lecture notes, (or videos) for studying pullback rings? I know definition; and a few facts about it (for example when it is Noetherian). Now I ...
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75 views

Monomials and initial ideals

I am working on two questions for my Commutative Algebra assignment and am struggling to finish them. $1.$ Let $S=K[x_1,...,x_n]$, $I\subset S$ an ideal and $<$ a term order. I first showed that ...
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54 views

Does every module over a commutative local ring direct sum of all its indecomposable submodules?

Let $R$ be a commutative local ring and $M$ be an $R$-module. To determine the structure of $M$, does it suffice to determine the structure of all indecomposable summands of $M$ and then say that $M$ ...
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66 views

A relation between max-spectrum and spectrum of a ring

For a commutative ring $R $ with identity we know that if the prime spectrum, the set of all prime ideal with Zariski topology, is noetherian then max spectrum, the set of all maximal ideal, is also ...
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46 views

Exercise 2.2 in Atiyah and Macdonald's Introduction to Commutative Algebra.

I am asked to prove that $(N:P) = Ann((N+P)/N)$, where $N,P$ are submodules of a module $M$, and $x\in (N:P) \Leftrightarrow xP \subset N$. I was thinking along the following lines: $$ x\in ...
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81 views

An inequality about the dimension of fiber

I am working on Problem 11.4.A of Vakil's notes: Let $X$ and $Y$ be two locally noetherian schemes, and $\pi:X \to Y$ is a morphism. $\pi(p)=q$. Then prove: $codim_Xp \leq ...
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32 views

Fractional ideals in the quotient field of Dedekind

Let $R$ be a Dedekind ring, $K$ its quotient field. If $J$ is a fractional $R$-ideal in $K$ then I want to show that $KJ=K$, so that it's a full $R$-lattice in $K$. Since $J$ is non-zero, we can ...
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53 views

Is it true that taking injective hull commutes with the tensor product?

Let $M$ and $N$ be two modules (can assume them to be finitely generated if need be) over the ring $A=k[x_0,...,x_n]$. Denote by $E(M)$ the injective hull of $M$. We work in the category of positively ...
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42 views

Radical of an ideal in $R [x]$

Let $\frak {I}$ be an ideal of $R[x]$, the polynomial ring over a commutative ring with identity $R$. Is it true that the radical of $\frak{I}$, the intersection of all prime ideals containing ...
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22 views

Locally presentable sheaves and the associated module functor

Let $R$ be a commutative ring. Any $R$-module has a presentation $R^{(J)}\rightarrow R^{(I)}\rightarrow M\rightarrow 0$. The associated module functor $M\mapsto \tilde M$ is exact and so preserves ...
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Spec $\mathbb Q[x]$ [duplicate]

I am trying to find Spec $\mathbb Q[x]$. Since $\mathbb Q$ is a field, the prime ideal is just generated by irreducible polynomial with coefficients in $\mathbb Q$. I know the case of $\mathbb R$, ...
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48 views

Codimension and height of prime ideals

Definition: Let $Z$ be an irreducible closed subset of $X$. Then the codimension $\textrm{codim} (Z,X)$ is the supremum of integers $n$ such that there exists a chain $$ Z = Z_0 < Z_1 < ...
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55 views

Idempotent direct summands of rings

I know that if an ideal $I$ is a direct summand of a ring $R$ then it is an idempotent ideal, i.e. $I^2=I$. My question concerns the rings all of whose idempotent ideals are direct summands. ...
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22 views

Torsion-free submodule of maximal rank

Let $A$ be a free $\mathbb{Q}[t_1^{\pm 1},\dots,t_\mu^{\pm 1}]$-module of rank $n$. If $M$ is a submodule of $A$, I know that it must be torsion free. Assuming the rank of $M$ is $n$ can I conclude ...
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(Atiyah) If $A \subseteq B$ and $B \setminus A$ is multiplicatively closed then $A$ is integrally closed in $B$ [duplicate]

I've tried proof by contradiction, with $y \in B\setminus A$ and considering an integral expression $y^n = a_{n-1} y^{n-1} +\dots + a_0$ of least degree (hence $a_0 \neq 0).$ Then $a_{n-1} y^{n-1} ...
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17 views

Finite number of maximal ideals above a maximal ideal in general integral extension [duplicate]

Let $A\subseteq B$ be an integral extension of integral domains and let $K$ and $L$ be the fields of fractions of $A$ and $B$ respectively. Assume that the field extension $L/K$ is Galois with finite ...
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56 views

Principal open sets in graded rings

I am interested in Prop II.2.5b in Hartshorne stating that if $D_+ (f) = \{p \in \textrm{Proj } B \mid f \notin p \}$ then there is a canonical homeomorphism $D_+ (f) \cong \textrm{Spec } B_{(f)}$ the ...
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34 views

$\textrm{Dim } S(Y) = 1 + \textrm{Dim } Y$

Let $k$ be algebraically closed, and $S = k[Y_0, ... , Y_n]$, and let $\mathscr Y$ be a variety in projective $n$ space, corresponding to the homogeneous prime ideal $\mathscr P$ of $S$. Let $U_0 = ...
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30 views

Every irreducible component of $Z(\mathfrak a)$ has dimension $\geq n-r$

I had a lot of trouble with this question and now have written down a solution I think is correct. Is there anything I have maybe overlooked in this proof or didn't do rigorously? If it is correct, ...
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Units in Semiperfect Skew Group Rings

Let $k$ be a field and $S$ the ring $k[[x_1,\ldots, x_n]]$. Let $G$ be a finite subgroup of $GL_n(k)$ that does not contain any nontrvial pseudo-reflections and such that $|G|$ is invertible in $k$. ...
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37 views

Radical ideals and prime decompositions for nonprincipal ideals

Fact. Let $R$ be a UFD. A principal ideal $ \left\langle a \right\rangle$ is radical iff every element in its prime decomposition has multiplicity one. Does this statement generalize in any way to ...
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36 views

Inverse of an isomorphism involving flat and finitely presented modules

I want to find the inverse of the map that appears in the black box. I have searched a lot and I could not find it. The books that I have come across display this proof technique without finding the ...
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61 views

Every nonzero prime ideal of $\mathcal {O_K}$ is maximal.

Theorem 1. If ring $B$ is an integral extension of ring $A$ and $P$ is prime ideal of $B$, then $P$ is maximal ideal of $B$ $\Leftrightarrow$ $A \cap P$ is maximal ideal of $A$. Theorem 2. If an ...
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28 views

When identifying $X$ with $\operatorname{mspec}k[X]$, how do we identify $\phi\colon X\to Y$?

Suppose $\phi\colon X\to Y$ is a morphism of affine varieties. It's common place to identify $X$ with the maximal ideals of $k[X]$, by $x\leftrightarrow M_x=\{f\in k[X]:f(x)=0\}$. What does this ...
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53 views

Field extension is étale implies polynomial is separable

Following Johnstone (Exercise 0.11), a ring homomorphism $f: A\rightarrow B$ is étale if for every nilpotent ideal $N\subseteq R$ of a ring $R$ and every diagram of ring homomorphisms there is a ...
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48 views

Isomorphism between ring of polynomial functions on unit hyperbola and Laurent polynomials [duplicate]

I want to prove that the ring $\mathbb{R}[X,Y]/(XY-1)$ of polynomial functions on the 'unit hyperbola' is isomorphic with the ring $\mathbb{R}[T,T^{-1}]\subset \mathbb{R}(T)$ of Laurent polynomials. ...
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36 views

Prove $I(\{(x,y,z) \in \mathbb{C}^3: x^2+y^2+z^2-1=0\}) = (x^2+y^2+z^2-1)\mathbb{C}[x,y,z] $

I got stuck in proving that $$I(\{(x,y,z) \in \mathbb{C}^3: x^2+y^2+z^2-1=0\}) = (x^2+y^2+z^2-1)\mathbb{C}[x,y,z]. $$ Let $X= \{(x,y,z) \in \mathbb{C}^3: x^2+y^2+z^2-1=0\}$, and $I(X) = \{ f \in ...
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48 views

Geometric intuition for $R[a^{-1}]$?

The ideal of polynomials vanishing over a point in an affine algebraic variety is maximal, and I think I understand the geometric intuition behind localizing it(s complement). But what about ...
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39 views

Completions and Localizations

I was wondering if the following is true: Let $(R,m,k)$ be a Noetherian local ring. If $R$ is a complete local ring (with respect to the $m$-adic topology), then $R_p$ is a complete local ring. ...
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45 views

Polynomial time algorithm for variety of an ideal

Is there a polynomial time algorithm to determine the variety of a zero dimensional ideal in $k[x_1,\ldots,x_n]$? Or is it a NP hard problem?