# Tagged Questions

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

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### How do I find the ideal $I+J$ and quotient $R/(I+J)$?

This is a homework problem: Consider the polynomial ring $R=\mathbb Z_2[x_0,x_1,\dots,x_n]$. Let $I=\langle x_0x_1\cdots x_n\rangle$ and $J=\langle x_0+x_1,x_0+x_2,\dots,x_0+x_n\rangle$. Find $I+J$...
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### A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. [closed]

I expect that the following result is true, but i can't prove it. A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. I need some help to prove this....
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### Zariski tangent vectors, dual numbers

Let $k$ be a field, $A$ be a Noetherian local $k$-algebra, $m$ its maximal ideal, and an isomorphism $i:A/m \to k$ . Let $v:m/m^2 \to k$ be a $k$-linear map (i.e. a Zariski tangent vector). I believe ...
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### Example of an monomial ideal that is weakly reverse lexicographic but not reverse lexicographic

We are looking at a paper titled "Generic Ideals and Moreno-Socias Conjecture" by Edith Aguirre, et al. In the paper they state that an ideal which is reverse lexicographic is also weakly ...
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### Residue field of the integral closure of a local ring in its field of fractions

When considering the discrete valuation rings contained in the rational functions field $R(F)$ of an irreducible plane projective curve $F \in \mathbb{P}^2(K)$ ($K$ algebraically closed), one can find ...
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### Rings of Krull dimension one

I have to write a monograph about commutative rings with Krull dimension $1$, but I can't find results, so I am looking foward for some references, and some results to search. Also, I would appreciate ...
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### On graded Artinian Gorenstein algebras

Let $k$ be a field and $R$ an $\mathbb{N}$-graded $k$-algebra that is graded-commutative. Assume that $\dim_k R<\infty$ and that $R$ is Gorenstein (i.e. the injective dimension of $R$ over itself ...
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### Motivation for localization as given in Eisenbud

Eisenbud writes that the affine ring $A(X-Y)$ is obtained from $A(X)$ by adjoining a multiplicative inverse of $f$, where $Y$ is the vanishing set of the function $f$. $A(X-Y)$ is the set of ...
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### grading of the tensor product

I have just had a look at http://therisingsea.org/notes/GradedModules.pdf to look up the grading of the tensor product of two graded modules over a graded ring (see page 10). And I am wondering, why ...
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### Let $\phi:A\to B$ be a ring homomorphism, $\phi^{*}:Y\to X$ the induced continuous map on $X=\mathrm{Spec}(A), Y=\mathrm{Spec}(B)$.

This is from Atiyah and MacDonald, Exercise 1.21, part iii). We let $Z=\mathrm{Spec}(R)=\{\mathfrak{p}\subset R\mid\mathfrak{p}\mathrm{\,a\,prime \,ideal}\}$ have the Zariski topology, i.e. with ...