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Finding if an ideal is the radical of another one

Let's suppose we have, in the ring $\mathbb{Z} [x,y,z,w,v]$, the following polynomials: $f=xw-yz$, $g=x^2z-y^3$, $h=yw^2-z^3$, $k=xz^2-y^2w$. The question is to prove that $I=(f,g,h,k)$ is the radical ...
WittyCatchphrase's user avatar
1 vote
1 answer
91 views

Quotient rings of integral quaternion rings

I'm having a hard time finding information about the quotient rings of the Lipschitz quaternions and the Hurwitz quaternions. The Lipschitz quaternions are defined as the quaternions with integral ...
A. Bailleul's user avatar
  • 1,303
1 vote
0 answers
80 views

Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?

Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
Alex's user avatar
  • 385
4 votes
0 answers
232 views

When $\langle u,v,w \rangle$ is a maximal ideal in $\mathbb{C}[x,y]$?

Let $u,v,w \in \mathbb{C}[x,y]$ and let $\langle u,v,w \rangle$ be the ideal generated by $u,v,w$. It is known that for two elements the following result holds: $\langle u,v \rangle$ is a maximal ...
user237522's user avatar
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1 vote
0 answers
52 views

Ideals of Laurent polynomial ring over matrix ring

Let $K$ be a field. Let $R=M_2(K)\langle x,x^{-1}\rangle$ be the ring obtained from the matrix ring $M_2(K)$ by adjoining two elements $x$ and $x^{-1}$ which are inverse to each other ($x$ and $x^{-1}$...
Ralle's user avatar
  • 471
1 vote
0 answers
129 views

Gelfand's representation on matrices: construct maximal ideal in matrix algebra

I would like to see a constructive proof (some algorithm?) of the following statement: Let $A_1, A_2, \dotsc ,A_k \in M_n(\mathbb C)$ be some commuting matrices, let $B$ be the commutative algebra (...
Zhang Yuhan's user avatar
1 vote
0 answers
216 views

Is the span of all nilpotent ideals also a nilpotent ideal?

Given a non-zero Lie algebra $\mathcal{L}$ over $\mathbb{C}$, we define $\mathcal{L}^2 = \big[\mathcal{L}, \mathcal{L} \big] = \big\{ [x, y]: x, y\in \mathcal{L} \big\}$, and for any $k\in\mathbb{N}$ ...
Sanae Kochiya's user avatar
3 votes
0 answers
93 views

An isomorphism problem for semigroups of ideals

An ideal of a semigroup $S$ (written multiplicatively) is a set $I \subseteq S$ such that $IS$ and $SI$ are both contained in $I$ (here, $XY$ means, for all $X, Y \subseteq S$, the setwise product of $...
Salvo Tringali's user avatar
0 votes
0 answers
57 views

Extracting implications in polynomial constraint system from Groebner basis

Given a Groebner basis for a system of polynomial constraints over $\mathbb{Q}$, are there any known methods for extracting the low degree factorable polynomials in the ideal generated by that basis? ...
PPenguin's user avatar
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1 vote
0 answers
65 views

Gorenstein property from initial ideal

My question is: If $I$ is a homogenous ideal of $S=K[x_1,\dots,x_n]$ and $in_{<}(I)$ is the initial ideal of $I$, with respect to a term order $<$ on $S$, then $S/I$ is Gorenstein if and only if ...
Chess's user avatar
  • 13
1 vote
1 answer
132 views

If $(f,g)$ and $(f,h)$ are maximal ideals, then $ag+bh=P(f)$ for some $a,b \in k, P(t) \in k[t]$?

Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$. Let $f,g,h \in k[x,y]$, $g \neq h$, satisfy the following two conditions: (1) $(f,g)$ is a maximal ideal of ...
user237522's user avatar
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0 answers
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Proof that the left ideal I of prime norm in maximal order can be written as I = ON + Oα

It seems there is a well-known fact that if $O$ is a maximal order in quaternion algebra $B$ and $I$ is a left $O-$ideal such that $nrd(I) = N$ is prime, then $I = ON + Oα$ with $gcd(N^2, nrd(α)) = N$....
student17's user avatar
0 votes
1 answer
134 views

$k(F_i)_{i=1}^{n}=k(G_j)_{j=1}^{m}$ iff there exist $a_i,b_j \in k$ such that $\langle F_i-a_i \rangle_{i=1}^{n} = \langle G_j-b_j \rangle_{j=1}^{m}$

Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$ and let $F_1,\ldots,F_n,G_1,\ldots,G_m \in \mathbb{C}[x,y]$, $n,m \in \mathbb{N}-\{0\}$. Claim: $\mathbb{C}(...
user237522's user avatar
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1 vote
1 answer
146 views

Ideals: If $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle$, then $\langle f_1-\lambda,f_2-\mu \rangle = \langle g_1-\delta,g_2-\epsilon \rangle$?

The following question appears in MSE without answers. Let $f_1,f_2,g_1,g_2 \in \mathbb{C}[x,y]-\mathbb{C}$. Assume that $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle \subsetneq \mathbb{C}[x,y]$, ...
user237522's user avatar
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0 votes
1 answer
144 views

$\mathbb{C}(u(x,y),v(x,y),f(x)+g(y))=\mathbb{C}(x,y)$ implies $\mathbb{C}(u(x,y),v(x,y))=\mathbb{C}(x,y)$?

The following question is a direct continuation of this question: Let $u,v \in \mathbb{C}[x,y]$. Assume that for every $f \in \mathbb{C}[x]$ and every $g \in \mathbb{C}[y]$ (excluding the cases where $...
user237522's user avatar
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0 votes
0 answers
98 views

Is that true that each ideal $I \subset k[x_1,\ldots,x_n]$ of finite $k$-codimension contains all monomials of sufficiently high degree?

Let $k$ be a field, and $I$ be an ideal in the $k$-algebra $k[x_1,\ldots,x_n]$ of all polynomials of $n$ variables. Suppose that $I$ has finite codimension over $k$, i.e. $$ \dim_{k} k[x_1,\ldots,x_n]/...
Sergiy Maksymenko's user avatar
2 votes
1 answer
359 views

What is the ideal of hypersurfaces singular at a given irreducible variety?

Let $X\subseteq \mathbb{P}^n$ be a closed irreducible subvariety, with vanishing ideal $I(X)\subseteq k[x_0,\ldots,x_n]$, where $k$ is the ground field, assumed to be algebraically closed. Let $F\in k[...
Jérémy Blanc's user avatar
6 votes
1 answer
266 views

The combinatorics of the Nullstellensatz for the variety of nilpotent matrices

Let $H_n$ denote the set of $n \times n$ nilpotent matrices with complex entries. The set $H_n$ may be regarded as an algebraic variety. Indeed, consider the polynomial ring $\mathbb{C}[A_{i,j} : 1 \...
Samuel Johnston's user avatar
3 votes
1 answer
222 views

Van der Waerden's classical paper on Hilbert polynomials and Bezout's theorem

Is it possible to obtain an electronic copy of Van der Waerden's "On Hilbert series of composition of ideals and generalisation of the Theorem of Bezout", Proceedings of the Koninklijke ...
pinaki's user avatar
  • 5,179
0 votes
0 answers
153 views

Product/intersection of two ideals

Let $I_{1}$ and $I_{2}$ be two ideals in polynomial ring $C[u,v,t]$, defined as $I_1 = \langle t-u^3, (v-u^5)^2\rangle$ and $I_2 = \langle u^{11} + v^{11} + t^{11} + 3\rangle$. Is there a method for ...
Ethan's user avatar
  • 1
1 vote
0 answers
40 views

Natural Density of Norms of ideals in a given ideal class

Some time ago, Landau proved the following formula for general number fields: $I_K(x)=U_Kx+O(x^{\delta})$, where $\delta=1-2/(1+[K:\mathbb{Q}])$, where $I_K(x)$ is the number of ideals with norm below ...
George Bentley's user avatar
0 votes
1 answer
202 views

Are zero dimensional ideals radical?

I have a question about Theorem 3.7.25. of Computational commutative algebra I by M. Kreuzer and L. Robbiano. Let $K$ be a perfect field, $I \subseteq K[x_1, \ldots, x_n]$, be a zero dimensional ...
Mairon's user avatar
  • 121
4 votes
0 answers
121 views

Which projections maintain irreducibility of the polynomial $x_0x_1 + x_2x_3 + \dots + x_{2m-2}x_{2m-1}$?

Let $q = x_0x_1 + x_2x_3 + \dots + x_{2m-2}x_{2m-1} \in \Bbb{C}[x_0, \dots, x_{2m-1}]$. The ambient space is $\Bbb{C}^{2m}$. Which are the polynomials $p \in \Bbb{C}[x_0, \dots, x_{2m-1}]$ such that ...
Varun Ramanathan's user avatar
2 votes
0 answers
70 views

Ideals by the polynomial with "shifted" variables like g(x,y,z,) g(y,z,u), g(z, u,v)

Are there any results related to properties of an ideal $I$ in $k[x_1,\ldots,x_n]$ generated by the polynomials $g(x_1,\ldots, x_m),\, g(x_2,\ldots, x_{m+1}), \ldots, g(x_{1+{n-m}},\ldots , x_{n})$? ...
olha's user avatar
  • 21
6 votes
1 answer
275 views

What is the right level of generality for $(R/a) \times (R/b) \cong (R/\gcd(a,b)) \times (R/\operatorname{lcm}(a,b))$?

Let $R$ be a principal ideal domain. Let $a$ and $b$ be two elements of $R$. Let $g$ be a greatest common divisor of $a$ and $b$, and let $\ell$ be a least common multiple of $a$ and $b$. (Of course, ...
darij grinberg's user avatar
1 vote
1 answer
113 views

How to compute the associated reduced ring for this finitely generated algebra?

Let $k$ be a field, $m$ be a positive integer and $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x,y]$. Let $B$ be the quotient ring $R/xR$. Then $B$ is the finitely generated $k$-...
Boris's user avatar
  • 569
0 votes
1 answer
188 views

Are the ideals in two $C^*$-algebras the same?

Let $V_{1}, V_{2}$ be the commuting isometries. By Wold decomposition theorem, we know that $V_{i}$ admits decomposition $$V_i \cong V^s_{i}\oplus V^{u}_{i},$$ where $V^{s}_{i}$ is the shift and $V^{u}...
Andy's user avatar
  • 139
7 votes
2 answers
483 views

Counterexample for Chvatal's conjecture in an infinite set

Let $X \neq \emptyset$ be a set. We say that ${\cal F} \subseteq {\cal P}(X)$ is a down-set if ${\cal F}$ is closed under taking subsets. Whenever $a \in X$, we let ${\cal F}_a = \{ S \in F : a \in S\}...
Dominic van der Zypen's user avatar
3 votes
1 answer
180 views

Ideals whose quotient rings have a certain property

There are some well-known properties of ideals which are equally well-known to correspond to properties of their respective quotient rings. For example: An ideal $p$ of a ring $R$ is prime iff $R/p$ ...
Cloudscape's user avatar
2 votes
1 answer
160 views

DCC on the powers of ideals

My apologies if this question is below the level of MO. I posted the same question in MS about a week ago without an answer so far. Let $R$ be a unital commutative ring. $R$ is called strongly $\pi$-...
Onur Oktay's user avatar
  • 2,283
3 votes
1 answer
239 views

Does the set of ideals, whose Jacobson radical & nilradicals coincide, form a sublattice?

This question might be below the level of MO, so apologies in advance. I posted the same question in MS about a week ago without an answer so far. Let $R$ be a unital commutative ring and $L(R)$ ...
Onur Oktay's user avatar
  • 2,283
2 votes
1 answer
138 views

Hadamard product of linear recurrences with umbral calculus

Let $R$ be a ring, $d_0, d_1, d_2, \dots \in R$ and $e_0, e_1, e_2, \dots \in R$ be linear recurrence sequences, such that $d_m = a_1 d_{m-1} + a_2 d_{m-2} + \dots + a_k d_{m-k}$ for $m \geq k$, $e_m ...
Oleksandr  Kulkov's user avatar
10 votes
2 answers
1k views

Is the ideal product presheaf a sheaf? Do we have any reasons to believe it will be / it won't?

My question is about one of those several concepts in algebraic geometry who everybody uses but nobody defines or introduces properly. Given a ringed space $(X,\mathcal{O}_X)$ and ideal sheaves $\...
Elías Guisado Villalgordo's user avatar
1 vote
2 answers
180 views

The quotient of an algebra with an ideal whose generators are decomposed as the product of irreducible elements

I would like to find reference for the following statement. I need it only in the particular case when $A=\mathcal{O}_{(\mathbb{C}^n, 0)}$ is the local algebra of holomorphic germs $(\mathbb{C}^n, 0) \...
Pintér Gergő's user avatar
5 votes
0 answers
132 views

Trying to prove a seemingly easy fact on ideals of ternary C*-algebras

Currently I'm reading the paper by Abadie and Ferraro titled Applications of ternary rings to $C^*$-algebras. Recall that a $C^{\ast}$-ternary ring is a complex Banach space $M$, equipped with a ...
Math Lover's user avatar
  • 1,105
0 votes
1 answer
149 views

Partial orders on downward closed sets [closed]

Let $P = (V, \sqsubseteq)$ be a partial order and $\mathfrak{D}(P)$ denote the class of downward-closed subsets of the partial order $P$ (i.e, the class of $A \subseteq V$ such that $y\in A \;\&\; ...
user65526's user avatar
  • 629
7 votes
1 answer
267 views

Does every infinite-dimensional Banach algebra contain an infinite-dimensional subalgebra with second-countable primitive ideal space?

Let $A$ be an infinite dimensional Banach algebra. Even if separable the primitive ideal space of $A$ need not be second-countable when endowed with the hull-kernel topology. Can we at least find an ...
Tomasz Kania's user avatar
  • 11.3k
3 votes
1 answer
321 views

A criterion for whether an ideal is contained in a principal ideal

Let $R$ be a ring and $I$ an ideal. I am interested under which conditions the following holds: Claim. Suppose that any two elements in $I$ have a non-trivial $\operatorname{gcd}$. Then $I$ is ...
mrtaurho's user avatar
  • 165
2 votes
0 answers
122 views

Quasi-ideals and Erdős conjecture on arithmetic progressions

Starting to read a book about algebraic geometry as well as the Wikipedia article on Erdős conjecture on arithmetic progressions, I came to think of what follows. Let $A$ be a set of positive integers,...
Sylvain JULIEN's user avatar
2 votes
1 answer
232 views

Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring

Question 1. Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I$ and $J$ be two ideals of $R$ such that $I$ is ...
darij grinberg's user avatar
4 votes
1 answer
235 views

$A$ is a commutative connective dg-algebra satisfying $H^0(A)=k$. Is it true that a dg ideal generated by elements of negative degree acyclic?

Let $k$ be a field of characteristic zero and $A$ be a graded commutative dg-algebra over $k$ with differential of degree $+1$ satisfying $H^0(A)=k, H^i(A)=0$ for $i<0$. Denote by $\mathcal J$ a dg ...
user avatar
2 votes
1 answer
231 views

Primitive ideals of minimal tensor product

Let $A$ and $B$ be $C^{\ast}-$ algebras and $A \otimes B$ denotes minimal(spatial) tensor product. Is there any classification of primitive ideals of $A \otimes B$? (I'm mainly interested in the ...
Math Lover's user avatar
  • 1,105
1 vote
1 answer
113 views

Symbolic power of an ideal associated to non-singular algebraic set

Let $Z\subset \mathbb P^n$ be a reduced non-singular algebraic set and $I$ denote the saturated homogeneous ideal of $Z$. I have seen the following result without proof: For all $ n\geq 1$, $I^{(n)}=(...
Cusp's user avatar
  • 1,703
9 votes
2 answers
1k views

Do relatively prime polynomials $f$ and $g$ in $k[x,y]$ generate an ideal of finite codimension?

Let $k$ be a field and $f,g \in k[x,y]$ be two non-constant polynomials in two variables. Is it true that the following conditions are equivalent: $f$ and $g$ are relatively prime in $k[x,y]$, in the ...
Sergiy Maksymenko's user avatar
1 vote
2 answers
477 views

Extension of the radical and radical of the extension of an ideal

If $A$ is a commutative ring, $I \subset A$ an ideal and $f:A \rightarrow B$ a ring homomorphism, then the extension of $I$, $I^e = \langle f(a): a \in I \rangle$ does not commute with the radical, I ...
Mario G's user avatar
  • 113
2 votes
1 answer
287 views

How can I prove this claim about splitting of prime ideals in real cyclotomic fields?

Let $L_k = \mathbb{Q}(\zeta_{2^k} + \zeta_{2^k}^{-1})$ be the maximal real subfield of the cyclotomic field of conductor $2^k, k \ge 2$ and $f_k(x)$ be the minimal polynomial of $\zeta_{2^k} + \zeta_{...
wyoumans's user avatar
  • 287
0 votes
0 answers
74 views

Sufficient conditions for $b\not\in I^2$ given that $b\in I$

Let $I$ be an $R$-ideal in a commutative algebra $B$ over a commutative ring $R.$ Given $b\in I$ I want to prove that $b\not \in I^2$. Are there any sufficient conditions for showing that $b\not\in I^...
Fallen Apart's user avatar
  • 1,615
1 vote
0 answers
99 views

When do the kernels of module homomorphisms between rings whose kernels contain a given fixed ideal contain every prime ideal over it?

$\DeclareMathOperator{\Hom}{Hom}$All our rings are commutative with unity and, if necessary, we can suppose that they are actually polynomial rings over a field in finitely many variables where the ...
Hvjurthuk's user avatar
  • 573
3 votes
1 answer
279 views

Is the kernel of an action of a Hopf algebra on an algebra a biideal?

I think, this must be simple, but I am not a specialist in this field, so excuse me. I asked this a week ago at MSE, but without success. S.Dascalescu, C.Nastasescu and S.Raianu define the action of a ...
Sergei Akbarov's user avatar
5 votes
1 answer
685 views

On the annihilator of a module

Question. Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module. Does there always exist an element $s\in M$ such that $\mathrm{Ann}(s)=\mathrm{Ann}(M)$? Remark. The annihilator of a ...
Chris 's user avatar
  • 303