# Tagged Questions

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

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### Integral closure of $R[x]$ in its field of fractions over R

I feel like this might have been discussed before but I couldn't find it so I apologise if this is a very common question. If $S$ is a ring and we have a subring $R$ and an element $x\in S$ ...
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### Geometric differences between $\operatorname{Spec}\mathbb{C}[x]/(x^2-x)$ and $\operatorname{Spec}\mathbb{C}[x]/(x^3-x^2)$

As far as I can tell, the topological spaces associated to the schemes in the title are both sets with two elements, with the discrete topology since both have prime ideals $(x)$ and $(x-1)$ which are ...
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Let $L$, $M$ and $N$ be $R$-modules. Then I know that there is a natural homomorphism from $Hom(L,M) \otimes N \to Hom(L,M \otimes N)$ defined by $f \otimes n \to \tilde{f}$ where $\tilde{f}(\ell)=f(\... 2answers 29 views ### Principal Ideal Domain and Factorization If$A$is a local domain such that each non-trivial ideal factors uniquely into primes then does it follow that$A$must be a principal ideal domain? 3answers 26 views ### Minimal ideal in a ring which is generated by an idempotent element. Let$R$be a commutative ring with unity and$M$be a minimal ideal of$R$such that$M = Re$where$e$is an idempotent element in$R$. Then$R = Re \oplus R(1-e) $I am not able to see, in order ... 2answers 46 views ### What are the conditions needed for two principal ideals of a ring to be isomorphic? Given a commutative ring$R$, and$p(x),q(x) \in R[x]$monic polynomials, under what conditions on$p(x)$and$q(x)$are the principal ideals$\langle p(x) \rangle$and$\langle q(x) \rangle$... 1answer 25 views ### Given a commutative ring$R$and a monic polynomial$p(x) \in R[x]$is$R[x]/\langle p(x) \rangle$always a finite integral extension of$R$? I suspect this to be true based on the fact that$p(x)$is monic, so it should be the case that$R[x]/\langle p(x) \rangle$is a finitely generated module over$R$, but I have no good reference for ... 1answer 34 views ### Minimal graded free resolution of the ideal$(x^3,xy^2,y^5)$I am looking for a detailed explanation of every step of the construction of a graded free resolution of the ideal$(x^3,xy^2,y^5) \subseteq S=K[x,y]$where$K$is an arbitrary field. I saw several ... 0answers 31 views ### on the proof of Theorem 2.11 of Ratliff's “On prime divisors of$I^n$,$n$large” Let us consider Ratliff's paper from 1975 entitled On prime divisors of$I^n$,$n$large, in particular the last statement of the first paragraph of the proof of Theorem 2.11. We have a Noetherian ... 2answers 118 views ### Minimal injective resolution of a module Let$R$be a commutative Noetherian ring and$M$an$R$-module. Let$0\rightarrow M\rightarrow E^{\bullet}$be a minimal injective resolution of$M$and$0\rightarrow M\rightarrow I^{\bullet}$be an ... 0answers 71 views ### A test problem about algebraic integers in complex field In a recent algebraic test, I meet this problem: Let R be the ring of algebraic integers in C, K is the field of algebraic numbers in C. Let a be an element of K such that the ring R[a] is ... 0answers 46 views ### Koszul complex: isomorphism between$K(a_1,\ldots, a_n;A) \simeq K(a_1;A) \otimes \cdots \otimes K(a_n;A)$Given$a_1,\dots,a_n\in A$, with$A$a suitable ring, my algebra teacher defined the Koszul complex associated to$a_1,\dots,a_n$with coefficients in$A$in this way: $$K(a_1,\dots,a_n;A):=\... 0answers 30 views ### Minimal injective resolutions isomorphism [closed] How can I prove that given an A-module M two injective resolutions of M are isomorphic as complexes? Thank you, have a nice day Asdrubale 1answer 49 views ### Is it \prod_{i \in \Delta } (S^{-1}A_{i}) \cong S^{-1} \prod_{i \in \Delta }A_{i} true? [closed] Suppose that A is a ring and S is a multiplicative closed subset of A. I wonder that do we have \prod_{i \in \Delta } (S^{-1}A_{i}) \cong S^{-1} \prod_{i \in \Delta }A_{i}? Can you give me ... 0answers 42 views ### translation of “Der kanonische Modul …” Do you know a note that is the translation of the following in English? J. HERZOG et al., "Der kanonische Modul eines Cohen-Macaulay-Rings," Lecture Notes in Mathematics No. 238, Springer-... 2answers 64 views ### Is a finite inverse limit of noetherian rings noetherian? Let \{A_i\} be an inverse system of (commutative, unital) Noetherian rings with a finite index set. Is \varprojlim A_i also a Noetherian ring? 1answer 49 views ### Example of a non-Kummer totally tamely ramified Galois extension Let A be a DVR with fraction field K, and let L be a totally tamely ramified finite Galois extension of K of degree e - ie, the integral closure B of A in L is a DVR with ramification ... 0answers 33 views ### Show that if f_{M}: L_{M} \to G_{M} is surjective for every maximal ideal M of R then f is surjective. Let f:L\to G be a homomorphism of modules over commutative ring R. Show that if f_{M}: L_{M} \to G_{M} is surjective for every maximal ideal M of R then f is surjective. 1answer 33 views ### Why is the Rees Algebra Noetherian if the underlying ring is? Let R be a commutative ring with 1, I \subset R a proper ideal. The Rees Algebra, with respect to I, is defined: R[It]= \bigoplus_{n=0}^\infty I^nt^n \subseteq R[t]. In many places I've read ... 1answer 59 views ### Obtain dimension of multivariate polynomial quotient ring? Let \mathbb{C}[z_1,z_2,...,z_n] be the ring of multivariate polynomials in complex variables z_1,z_2,...,z_n with complex coefficients. This ring is spanned by the countably infinite basis of ... 1answer 40 views ### Hartshorne Prop I.4.3 Proof \textit{The proposition}: On any variety Y, there is a base for the topology consisting of open affine sets. \textit{The proof}: Assume Y is quasi-affine in \mathbb{A}^n and let Z=\... 0answers 21 views ### If N is finitely generated, then (L :_{R} N)^{e}= (S^{-1}L :_{S^{-1}R} S^{-1}N) [duplicate] I have question in this lemma. Please help me explain it more. Let L, N be submodules of a module M over a commutative ring R and let S be a multiplicatively closed subset of R. If N is ... 1answer 42 views ### Prime ideals in a quotient of a DVR Suppose R is a DVR. So R has two prime ideals - (0) and (p) (p the uniformizer of the maximal ideal). All other ideals in R are powers of (p), i.e. of the form (p^k), k\geq 2. I'm ... 2answers 579 views ### Vanishing of a certain Tor I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ... 1answer 829 views ### Tensor product of reduced k-algebras must be reduced? Let A, B be two reduced k-algebras. Then if an element of the form$$\sum a_{i}\otimes b_{j}$$is nilpotent, we can compose it with any$k$-homomorphism$f$from$A$to$k$to get a homomorphism ... 0answers 60 views ### Prime ideal of a polynomial ring in 6 variables Let$k$be a field and$k[x_1,x_2,x_3,y_1,y_2,y_3]$a polynomial ring in 6 variables over$k$. How to prove that the ideal$(x_1y_2-x_2y_1,x_2y_3-x_3y_2,x_3y_1-x_1y_3)$is prime in$k[x_1,x_2,x_3,y_1,...
Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every field extension $L/\Bbbk$, and reduced if its underlying ring is reduced. Separable always implied reduced, and I found ...