Tagged Questions

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Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $K[X_1,X_2,X_3,X_4]$

Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements in the ring $R=K[X_1,X_2,X_3,X_4]$?
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When is a local, reduced, (commutative) ring an integral domain?

Question I am wondering whether or not it is true that if $A$ is a reduced ring, then is it the case that the localization of $A$ at any of its prime ideals is an integral domain? Discussion ...
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Orthogonal idempotents from disjoint union in $\text{Spec}(A)$ [duplicate]

Let $A$ be a commutative ring with unity. Suppose that $X=\text{Spec}(A)$ is a disjoint union $X_1\cup X_2$ of topological spaces. Show that $A$ has a pair of orthogonal idempotents $e_1,e_2$ ...
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Show that quotient ring of a $\Bbb C$-algebra by a maximal ideal is isomorphic to $\mathbb{C}$.

Let $R = \mathbb{C}[x_1,...,x_n]/I$ be a quotient of a polynomial ring over $\mathbb{C}$, and let $M$ be a maximal ideal of $R$. How do I show that quotient ring $R/M$ is isomorphic to ...
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Generating set of a free module is its basis

Let $F \cong R^d$ be a free R-module, where R is Noetherian. Let $Y$ be a generating set for $F$ with $|Y| \leqslant d$. Show that $Y$ is a basis for $F$ and $|Y|=d$. Thought that this must be like ...
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Epimorphism from arbitrary module to a free module

Let $F$ be a free module and let $f:M \rightarrow F$ be an epimorphism from a module $M$ onto $F$. Show that there exists a homomorphism $h:F \rightarrow M$ such that $f \circ h = id_F$ and deduce ...
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Finitely generated modules over a Noetherian ring are Noetherian

I'm trying to prove that if the ring $R$ is Noetherian then every finitely generated $R$-module is Noetherian. First of all, it is known that every module is a homomorphic image of a free module, ...
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Suppose that for every nonzero $R$-ideal $I$ and element $a \in I$ there exists a unique $R$-ideal $J$ such that $IJ=(a)$. Then $R$ is Noetherian.

Suppose that for every nonzero $R$-ideal $I$ and element $a \in I$ there exists a unique $R$-ideal $J$ such that $IJ=(a)$. Then $R$ is Noetherian. I'm having trouble proving this. To note, $R$ is ...
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When does there exist a commutative ring $C$ that contains rings $A$ and $B$ as a subring?

The statement I'm trying to prove is the following: Let $A$ and $B$ be commutative rings, both of characteristic $0$. Then there exists a commutative ring $C$ that contains both $A$ and $B$ as ...
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Atiyah Macdonald Chapter 3 Problem 23 Part ii)

I am really confused about Atiyah Macdonald chapter 3 problem 23 part ii) The set up: Let $A$ be a ring and $X=\text{Spec}(A)$ be the set of prime ideals of $A$ with the Zariski topology. Let $U$ be ...
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Given a f.g. module with this property

Give a finitely generated $R-$module with at least one submodule which has a infinite generator set ($R$ need not to be Noetherian).
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Elementary method for finding $I(Y)$ for the curve $Y$ defined parametrically by $x=t^{3}$, $y=t^{4}$, $z=t^{5}$

In order to motivate some of the theory we will be learning in a computational commutative algebra course, my professor assigned a number of computational problems that are [seemingly] quite difficult ...
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Atiyah-MacDonald help with exercise 5.10

This is an exercise from Atiyah-MacDonald, if someone can give an idea on how to prove that $a)\Rightarrow b)$: Let $f:A\rightarrow B$ a ring homomorphism. a) ...
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Atiyah-MacDonald 5.10

The problem says: Let $f:A\rightarrow B$ be a ring homomorphism and let $f^{*}:\operatorname{Spec}(B)\rightarrow \operatorname{Spec}(A)$ be the mapping associated with $f$. And then comes the ...
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Find a structure for Hilbert series

Assume $I$ is a monomial ideal in $R=k[x,y]$, $I=\langle m_1,\ldots,m_t\rangle;\ m_i= x^{a_i}y^{b_i}$. I want to find a structure for Hilbert series $R/I$ which depends on $a_i$ and $b_i$ such that ...
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Product of (strongly) stable ideals and lexsegment ideals

(1) Is the product of lexsegment ideals again a lexsegment ideal? (2) Is the product of (strongly) stable ideals again (strongly) stable? I know that both of them are false and I can find examples ...
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Map between two direct limits

Let $\{ M_i, ϕ_j^i\}_{i\in I}$ be a direct system of $R$-modules over a direct index set $I$. Show that there exists a direct system $\{P_i,\psi_j^i\}_{i\in I}$ of projective $R$-modules and a ...
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Prime ideal in the ring of polynomials

I'm trying to do the following: Let $R = K[X,Y,Z]$ and $\mathfrak{p}$ = $(X+Y,Z^{2}-X)$. Show that $\mathfrak{p}$ is prime and find the transcendence degree of $R/\mathfrak{p}$. If I prove ...
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Krull dimension of a $\mathbb Q$-algebra

I'm trying to find the Krull dimension of $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})$. My professor said that I have to consider that $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})$ is a $\mathbb{Q}$-algebra but I ...
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The h-vector of a simplicial complex

Let $S$ be a polynomial ring over a field. I want to find an ideal $I\subseteq S$ such that $(1,2,3,1,1,1)$ is the $h$-vector of $S/I$. We have a relation between $f$-vector and $h$-vector and ...
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Maximal subrings of $\mathbb{Q}$

Consider the sets $$\mathbb{Q}_p= \left\{ \frac{a}{b} \in \mathbb{Q}\mathbin{\Large\mid} b \notin (p) \right\}$$ Are these all the maximal subrings of the rationals?
Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$-module. Take $N$ a submodule of $M$ and let $P$ be a divisible $R$-module. Prove that any homomorphism $f: N \rightarrow P$ ...