All questions with tag [math: commutative-algebra]


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 17:47:17

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 17:47:06

Is $f$ reduced if and only if the derivations $\gcd(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})=1$ under some conditions?

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 17:43:48

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 17:19:56

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 17:19:48

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 17:19:15

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 17:18:57

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 17:18:05

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 17:17:50

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 17:14:32

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 & \text{ for } \{a,b\}\subset...
2022-07-25 17:08:39

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 $\operatorname{grade}(I,M)>\operatorname{height} I$?
2022-07-25 16:57:50

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 16:56:51

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 16:42:37

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 16:37:06

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 $n>0$. Then $M$ is Noetherian if and only if $M$ is Artinian Do you have any idea how to solve this?
2022-07-25 13:31:37

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 13:30:39

$S^{-1}M \cong S^{-1}N$ 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 13:21:26

$N$ 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 13:19:42

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 13:14:43