# Tagged Questions

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

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### software with a routine for the vanishing ideal of a finite set of points

I am looking for an algebraic software package that provides a routine that computes the vanishing ideal of a finite set of points. So far i am working with Macaulay2 but i have not been able to find ...
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### The intersection of two minimal prime ideals.

Let $A$ be a reduced commutative ring (that is, $A$ has no nontrivial nilpotents) and $P_1$, $P_2$ two minimal prime ideals of $A$. Is it true that the intersection of $P_1$ and $P_2$ is zero? It ...
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### When $Rx = Re$ and $e^2 =e$

Let $R$ be a commutative ring with identity. Suppose $x , e \in R$ with $Rx = Re \mbox{ and } e^2 = e$. what is the best thing that we can say about $x$?
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### Is injectivity of algebras preserved by tensor products?

Suppose $R' \subset R$, $S'\subset S$ are inclusion of $k$-algebras. Does it hold that $R'\otimes_kS' \rightarrow R \otimes_k S$ is injective ? I know there're counterexamples for modules, but why ...
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### Derived category of certain ring

I'm interested in the structure of $D^b(R)$, where $R=k[x]/(x^n)$. How one can describe this category? What is the list of indecomposable objects in this category?
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### Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
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### Submodules of a Noetherian module are finite intersections of irreducible submodules [duplicate]

If $M$ is a Noetherian $R$-module then every submodule of $M$ is a finite intersection of irreducible submodules. Please show me the way how to get the proof of this statement.
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### Is $\mathbb{C}[[x]] \simeq \mathbb{C}[x]_{(x)}$?

Let $\mathbb{C}[x]$ be the ring of polynomials and $\mathbb{C}[[x]]$ the formal power series. Is $\mathbb{C}[[x]] \simeq \mathbb{C}[x]_{(x)}$? Is it true? Is there a geometric interpretation of this ...
### Knots and reducible spectra $\mathbb{C}[\![x,y]\!]/I$
Let $I=(y^2-x^3-x^2)$ be an ideal of $\mathbb{C}[x,y]$. I don't know why $\operatorname{Spec}(\mathbb{C}[x,y]/I)$ is irreducible but $\operatorname{Spec}(\mathbb{C}[\![x,y]\!]/I)$ is reducible. Do you ...