Questions tagged [ideals]
The ideals tag has no usage guidance.
146
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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 ...
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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 ...
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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 ...
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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 $R$ ...
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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 ...
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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 $x^{-1}$...
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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 (...
- 11.4k
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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 ...
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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 $k\in\mathbb{N}$ ...
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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 $...
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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?
...
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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 ...
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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 ...
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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 $gcd(N^2, nrd(α)) = N$....
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$k(F_i)_{i=1}^{n}=k(G_j)_{j=1}^{m}$ 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}(...
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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 $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle \subsetneq \mathbb{C}[x,y]$,
...
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$\mathbb{C}(u(x,y),v(x,y),f(x)+g(y))=\mathbb{C}(x,y)$ 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 $...
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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]/...
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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[...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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})$? ...
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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, ...
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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$-...
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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}...
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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\}...
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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) \...
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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 $R/p$ ...
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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 $\pi$-...
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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 $L(R)$ ...
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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 ...
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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 $\...
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Cellular and primary binomial ideals
Let $I \subseteq \mathbb{K}[x_1, \dots, x_n]$ be an ideal of a polynomial ring over a field $\mathbb{K}$.
$I$ is called cellular if every variable $x_i$, with $i=1, \dots, n$, is either a ...
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Existence of maximal analytic P-ideal
An ideal $\mathcal{I}$ on the positive integers $\mathbf{N}$ is a P-ideal if for every sequence $(A_n)$ of sets in $\mathcal{I}$ there exists $A \in \mathcal{I}$ such that $A_n\setminus A$ is finite ...
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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 ...
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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 \;\&\; ...
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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 ...
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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 ...
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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,...
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$A$ 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 ...
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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 ...
- 30.3k
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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 ...
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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 ...
- 2,524
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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)}=(...
- 1,703
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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_{...
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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^...
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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 ...
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