# Tagged Questions

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### generators of an ideal

I've been thinking about this exercise but I can't get the solution. In $\mathbb{R}^3$ , I consider the usual axis: $l_1=\{ x_1=x_2=0 \}$, $l_2=\{x_1=x_3=0\}$ and $l_3=\{ x_2=x_3=0 \}$. Calculate ...
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I'd really appreciate if someone could help me. The problem is the following: If $\psi_1,...,\psi_m \in k[x_1,\dots,x_n]$ and consider the $k$-algebra homomorphism: ...
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### Noetherian ring of Krull dimension $0$

I've found this claim: Let $A$ be a Noetherian ring of Krull dimension $0$ . Then $A$ is a field or it has a finite number of prime ideals. Why is this true ?
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### prime ideals contains comaximal

Let $R$ be a commutative ring with unity 1 and $I$, $J$ and $P$ ideals in $R$ show that if every prime ideal of $R$ contains either $I$ or $J$ ,but not both then $I$ and $J$ are comaximal ...
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### Vanishing polynomials

Let $K$ be a field and $V$ be the set of points $(t^3,t^4,t^5)$ where $t$ is in $K$. Set $I=(Y^2-XZ,Z^2-X^2Y,X^3-YZ)$. Show that $I$ is a subset of $A$, where $A$ is the set of polynomials which ...
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### Degree of extension is equal to linear combination of prime factor multiplicities with prime factor index coefficients in Dedekind domains

I'm working on the following problem... Suppose that $A$ is a Dedekind domain with fraction field $K$. $L/K$ is a finite separable extension of $A$ of degree $n$, and $B$ is the integral closure ...
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### Every radical ideal in a Noetherian ring is a finite intersection of primes

Prove that if $I$ is a radical ideal and $ab\in I$, then $I=rad(I+(a))\cap rad(I+(b))$. Deduce that every radical ideal in a Noetherian ring is a finite intersection of primes. I've done the ...
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### Localisation and prime ideals

If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$. Prove that $Spec(A_f)=(\mathfrak{V}((f)))^c$. Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in Spec(A): P \supset (f)\}$ ...
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### Irreducible components of $Spec(A)$

A topological space $X$ is called irreducible if given $A_{1}, A_{2}$ open sets $\neq \emptyset$ then $A_{1} \cap A_{2} \neq \emptyset$. The maximal irreducible topological subspaces of $X$ are ...
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### Primary descomposition of ideals

I'd appreciate if someone could help me a bit with this problem. Considering $\mathfrak{p}=(x,y), \mathfrak{q}=(x,z)$ and $\mathfrak{m}=(x,y,z)$ ideals in $k[x,y,z], k$ field. Is ...
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### Let $R$ be a PID and $I$ is a non zero proper ideal of $R$. show that if $R/I$ has no nonzero zerodivisor, then it is a field. [closed]

Let $R$ be a PID and let $I$ be a non-zero proper ideal of $R$. Show that if $R/I$ has no non-zero zerodivisor, then it is a field.
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### Need an explanation for homomorphism in commutative algebra

I'm self-learning commutative algebra following "Introduction to Commutative Algebra". When dealing with concepts like "contraction" and "extension", some exercises in this book don't specify which ...
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### Is the ideal $I=(x_1 x_5 - x_2 x_4 , x_1 x_6 - x_3 x_4)$ of $k[x_1,…,x_6]$ a radical ideal? Is it a prime ideal?

Is the ideal $I=(x_1 x_5 - x_2 x_4 , x_1 x_6 - x_3 x_4)$ of $k[x_1,...,x_6]$ a radical ideal? Is it a prime ideal? thanks
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### Finitely many prime ideals lying over $\mathfrak{p}$

Let $A$ be a commutative ring with identity and $B$ a finitely generated $A$-algebra that is integral over $A$. If $\mathfrak{p}$ is a prime ideal of $A$, there are finitely many prime ideals $P$ ...
### Necessary and sufficient condition for $r(\mathfrak a)$ to be prime
As we know, $$\mathfrak a~\text{is a primary ideal}\Rightarrow r(\mathfrak a)~\text{is a prime ideal}.$$ But $r(\mathfrak a)$ may not be a prime ideal if $\mathfrak a$ isn't a primary ideal. ...
Let $R = \mathbb{Z}_{(2)}$ be the localization of $\mathbb{Z}$ at the prime ideal generated by $2$ in $\mathbb{Z}$. Then prove that the ideal generated by $(2x-1)$ is maximal in $R[x]$. Otherwise ...