# Tagged Questions

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

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### 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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### Dimension localization [closed]

Let $A$ be the localization of $\mathbb Z[x, y]$ in the ideal $(5, x−1, y+2)$ and $B = A/(x^2+y^2+4y−3x+6)$. Calculate the dimensions of $A$ and $B$ and study if they are regular rings.
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### Any ideal is an extended one

It is true for any commutative rings $S$ and $T$ with $1$ and any ring homomorphism $f:S\to T$ that the set $E$ of extended ideals in $T$ equals $\{J\mid J^{ce}=J\}$. In fact, if an ideal $J$ of $T$ ...
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### Sum of ideal sheaves commutes with taking global sections

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ effective divisors intersecting each other at finitely many points. Is it true that ...
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### Extension of idempotent ideals

Let $R$ be a Noetherian commutative ring with $1$. If $R[[x]]$ denotes the ring of formal power series over $R$ and $I$ is an idempotent ideal of $R$ I want to know whether the extension of $I$ in ...
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### Computing Krull dimension of $\mathbb{Z}[X_1,\ldots,X_n]/I$ [closed]

Let $I$ be an ideal of $\mathbb{Z}[X_1,\ldots,X_n]$. How does one compute the Krull dimension of $\mathbb{Z}[X_1,\ldots,X_n]/I$? Are there any general methods? Or methods which work in special cases?
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### Let $(R,M)$ be a local ring. Suppose that $R$ is noetherian and let $I,J \unlhd R$ such that $J \subseteq I$. Prove that the following are equivalent.

Let $R$ be a local ring with maximal ideal $M$. Suppose that $R$ is noetherian and let $I,J$ be ideals of $R$ such that $J \subseteq I$. Consider the following statements: 1) Every minimal set of ...
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### Is $\mathbb{R}[x,y,z]/(x^2+y^2+z^2)$ a UFD?

As the title says, I am curious as to whether $A =\mathbb{R}[x,y,z]/(x^2+y^2+z^2)$ is a UFD. I believe the answer is yes. A thought I had was to apply Nagata's criterion, say by localizing ...
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### Commutative algebra text that solely contains 200+ exercises

I am looking for a textbook that I came across awhile ago that I have been unable to find for the last week or so of periodic searching. The textbook had nothing other than, from memory 247 exercises ...
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This is Exercise 3c. from Chapter 9, Section 7 of Ideals, Varieties, and Algorithms by Cox et al. Given $I=\langle xy, xz+z(y^2-z^2)\rangle$, prove that $I=\langle x, z(y^2-z^2)\rangle \cap ... 1answer 57 views ### Does$S = R \cap K$of a field extension$K \subseteq L = Q(R)$satisfy$Q(S) = K$? If$K$is finite field, then one can easily show that there is no proper subring$R$with$Q(R) = K$, where$Q(R)$is the field of fractions of$R$. As a consequence, algebraic extensions$K$of ... 0answers 118 views ### If$A$is the ring of continuous functions on a genus$g$surface, can the genus of$X$be seen by simple algebra in$A$? [migrated] I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ... 0answers 21 views ### Locally presentable sheaves and the associated module functor Let$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$is exact and so preserves ... 0answers 16 views ### Counter example that Artinian k-algebra is not finite k-vector space [duplicate] Let k be a field, A be a k-algebra. If A is not a finitely generated k-algebra,then the following two conditions are NOT equivalent: (i) A is Artinian; (ii) A is a finitely k-algebra, i.e. A is ... 2answers 55 views ### If$f: A\to B$is faithfully flat and$B$is an Artinian ring then$A$is also Artinian. Let$f : A → B$be a map of rings. The map$f$is called faithfully flat if$B$is flat$A$-module ($B$is$A$-module w.r.t. multiplication defined by$ab := f(a)b$) and if for any$A$-module$M, M ...
What is the Krull dimension of $B = A[x,y,z]/\langle x^2y + x + 1, y^3 + 2z + 1 \rangle$ given $A$ is a Noetherian, commutative ring? (Assuming that all coefficients are non zero in $A$)