# All questions with tag [math: commutative-algebra]

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## Is the ideal $(X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ prime?

Consider the excellent $(f = X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ in the polynomial ring $k[X_0,\ldots, X_n]$. Is this a prime perfect? If so, what is its elevation? I'm stuck attempting to show that $f$ is irreducible.
2022-07-25 20:47:17
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## How to find a finite set of generators for $I \subset k[x_1, ..., x_n]$

Suppose that we have a set of polynomials $f_1, ..., f_m \in C = k[x_1, ..., x_n]$ where $k$ is an algebraically closed field, and let $V$ denote the set on which they all simultaneously vanish. Define $I(V)$ to be the ideal of polynomials in $C$ which vanish on all of $V$. In general, $I(V) = \text{rad}(f_1, ..., f_m)$ by Hilbert's nullstellens...
2022-07-25 20:47:06
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I have encountered the following problem. I have no ideas to prove it or disprove it. Suppose that $f\in \mathbb{C}[x,y]$, $f(0,0)=0$, $\frac{\partial f}{\partial x}(0,0)=0, \frac{\partial f}{\partial y}(0,0)=0$. Suppose $f$ is square-free—for simplicity we may first assume that $f$ is irreducible. Then $\frac{\partial f}{\partial x}$ and $\... 2022-07-25 20:43:48 5 ## Proving that surjective endomorphisms of Noetherian modules are isomorphisms and a semi-simple and noetherian module is artinian. I am changing for my Rings and also Modules test and also am stuck on the adhering to 2 questions:$1.$Let$M$be a noetherian component and also$ \ f : M \rightarrow M \ $a surjective homomorphism. Show that$f : M \rightarrow M $is an isomorphism.$2$. Show that if a semi - straightforward component is noetherian after that it is artin... 2022-07-25 20:19:56 2 ## Spectrum of finite$k$-algebras Let$k$be an area and also$A$be a limited$k$- algebra. Just how does one promptly see that$Spec(A)$is a limited set? Better, is it real that the cardinality of$Spec(A)$amounts to$dim_k(A)$? 2022-07-25 20:19:48 1 ## Is inclusion of a prime ideal into a different prime ideal possible? Let$A$be a commutative ring with identification. Allow$p, q$be 2 distinctive prime perfects. Is it feasible that$p \subseteq q$? 2022-07-25 20:19:15 1 ## Proof that the$d$-th powers generate the$d$-th symmetric power of a vector space Let$V$be a$\mathbb{C}$-vector space of finite dimension. Denote its$d$-th symmetric power by$V^{\odot d}$. I am looking for a proof that$V^{\odot d}$is generated by the elements$v^{\odot d}$for$v\in V$. A different way to look at it is the following: Consider the polynomial ring$R=\mathbb{C}[x_1,\ldots,x_n]$and$f$a homogeneous pol... 2022-07-25 20:18:57 1 ## Is the image of a tensor product equal to the tensor product of the images? Let$S$be a commutative ring with unity, and also allow$A,B,A',B'$be$S$- components. If$\phi:A\rightarrow A'$and also$\psi:B\rightarrow B'$are$S$- component homomorphisms, is it real that $$\operatorname{im}(\phi\otimes\psi)=\operatorname{im}(\phi)\otimes_S \operatorname{im} (\psi)?$$ 2022-07-25 20:18:05 1 ## Generating set for sum of two ideals Suppose there are two ideals$I,J \in \mathbb{C}[x_1,\dots,x_k]$and two sets of generating polynomials$\langle f_1, \dots, f_s\rangle$,$\langle g_1, \dots, g_t\rangle$. Now I want to describe$I + J$with a set of generating polynomials. Is $$\langle f_1, \dots, f_s, g_1, \dots, g_t\rangle$$ a valid generating system? 2022-07-25 20:17:50 1 ## What is the coproduct of fields, when it exists? This is a slightly more advanced version of . Let$\textbf{CRing}$be the category of commutative rings with unit. Let$\textbf{Dom}$be the category of integral domains – by which I mean a non-trivial commutative ring with unit such that the zero ideal is prime. Let$\textbf{Fld}$be the category of fields – by which I mean an integral domain s... 2022-07-25 20:14:32 1 ## Is the additive semigroup of natural numbers the multiplicative semigroup of a ring?$\mathbb N$will denote the set$\{0,1,2,\ldots\}.$The semigroup$(\mathbb N,+)$doesn't have a zero element.$\mathbb N^0$will denote the semigroup$\mathbb N$with zero adjoined, that is the set$\mathbb N\cup\{\star\},\;\star\not\in\mathbb N,$with the operation$+_0$defined by $$a+_0b=\begin{cases}a+b &amp; \text{ for } \{a,b\}\subset... 2022-07-25 20:08:39 1 ## Example where \operatorname{grade}(I,M)>\operatorname{height} I Let I be an ideal of a noetherian ring R and let M be a finite R-module. We need to show if I is generated by n elements, then \operatorname{grade}(I,M)\le n. Could any one give an example where \operatorname{grade}(I,M)&gt;\operatorname{height} I? 2022-07-25 19:57:50 1 ## How to show that the coordinate ring of a finite set of points in projective space is Cohen-Macaulay? Could any person offer me a tip just how to show this one: Let V be a limited set of factors in projective room. Just how to show that the coordinate ring of V is Cohen - Macaulay? 2022-07-25 19:56:51 0 ## Koszul Complex Homology I'm attempting to understand Eisenbud's proof that: If x_1,x_2,\ldots,x_i is an M-sequence, then H^i(M\otimes K(x_1,...,x_n))=((x_1,\ldots,x_i)M:(x_1,\ldots,x_n))/(x_1,\ldots,x_i)M. Here M-sequence means M-regular, i.e. (x_1,\ldots,x_i)M\neq M and for each j \in {1,\ldots,i}, x_j is not a zero divisor on M/(x_1,\ldots,x_{j-1}... 2022-07-25 19:42:37 1 ## Why is this ideal projective but not free? Let R=\mathbb{Z}[\sqrt{-5}] and also I=(2,1+\sqrt{-5}). Just how can I confirm that I is projective yet not free? 2022-07-25 19:37:06 2 ## Artinian if and only if Noetherian Let R be a ring (commutative, with identity), m a maximal ideal and M an R-module. Suppose m^nM=0 for some n&gt;0. Then M is Noetherian if and only if M is Artinian Do you have any idea how to solve this? 2022-07-25 16:31:37 2 ## Isomorphism between two localizations I am doing exercise 3.23 in Atiyah Macdonald and in the first part of the problem they ask to show that the ring A_f = S^{-1}A where S = \{1,f,f^2 \ldots \} depends only on the choice of the basic open set X_f and not on f. For reference,$$X_f \stackrel{\text{def}}{\equiv} \{P \in \operatorname{Spec}(A) : f \notin P\}.$$I interpret thi... 2022-07-25 16:30:39 1 ##$S^{-1}M \cong S^{-1}N$does not imply$M \cong N$Let$M, N$be$A$-modules, where$A$is a commutative ring with identity. Let$S$be a multiplicative subset of$A$that contains no zero divisors and contains the identity of$A$. I am looking for a counterexample to the statement$S^{-1}M \cong S^{-1}N \Rightarrow M \cong N$. Thanks. 2022-07-25 16:21:26 3 ##$N$submodule of$M$and$N \cong M$does not necessarily imply that$M=N$Let$M, N$be$A$-modules with$A$being a commutative ring. Suppose that$N$is a submodule of$M$and also that$N$is isomorphic to$M$. According to my understanding this does not necessarily imply that$M=N$. Is this statement accurate? If yes, at what kind of cases do we have this phenomenon? 2022-07-25 16:19:42 0 ## What is the injective hull of$\mathbb{C}(x,y)/\mathbb{C}[x,y]$? What is the injective hull of$\mathbb{C}(x,y)/\mathbb{C}[x,y]$as a$\mathbb{C}[x,y]\$ - component? Is it isomorphic to any kind of acquainted component?
2022-07-25 16:14:43