Questions tagged [ideals]
The ideals tag has no usage guidance.
146
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The number of ideals in a ring
Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here.
Let $R$ be a finite commutative ring with identity. Under what conditions the number of ...
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2
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Distribution relation in the Euler system of Heegner points
I am trying to understand the details behind the so-called "distribution relations" between Heegner points on the modular curve $X_0(N)$, as given (for instance) in Gross's paper Kolyvagin's work on ...
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1
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Inverse of the Structure Theorem for Finitely Generated Modules over PID
We know that for a PID $R$, any finitely generated module is of the form $\frac{R}{(a_1)} \oplus \dots \oplus \frac{R}{(a_s)} $.
I was wondering if the converse of this statement is true, that is, is ...
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Can an intersection of ideals in a Noetherian ring be replaced by a countable intersection?
Let $(R,\frak m)$ be a Noetherian local ring, and let $X$ be a set of ideals in $R$. Assume $\bigcap_{I \in X} I = 0$. Is there some sequence $\{I_n\}_{n \in \mathbb N}$, with $I_n \in X$ for all $n$...
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2
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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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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 ...
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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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Ideals invariant under ring automorphisms
I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:
$I$ is generated by two homogeneous elements;
$I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...
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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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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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Lifting flat modules over ring quotients
Let $R$ be a commutative ring, $I$ its ideal, and $\bar{R}=R/I$. For which flat $\bar{R}$-modules $\bar{F}$ is there a flat $R$-module $F$ such that $F \otimes_R R/I \simeq \bar{F}$?
By Lazard's ...
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Intersections of ideals in polynomial rings with countably many variables
Fix a field $k$ and let $R = k[x_1,x_2,\ldots]$. Say that an ideal $I \subset R$ is generated in finite degree if there exists a generating set $S$ for $I$ (possibly infinite) and an integer $n$ such ...
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Maximal subideal of an ideal
For a commutative ring $R$ with unity, I am looking for an equivalent condition for an ideal $T$ to have the property that $T$ contains a unique maximal proper subideal, equivalently, the sum of ...
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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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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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On GCD and LCM of elements in integral domain with Krull-dimension 1
Let $R$ be an integral domain with Krull dimension $1$. If $0\ne a \in R$ is such that for every $b \in R$ , the ideal $Ra \cap Rb$ is principal , then is it true that for every $b\in R$, the ideal $...
user111524
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Is every 2-sided ideal in a C*-algebra hereditary?
If $A$ is a C*-algebra, we say that a subset $I\subseteq A$ is hereditary if
$$
0\leq x \leq y \in I \Rightarrow x\in I.
$$
It is is well known that closed 2-sided ideals are hereditary.
Would it ...
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Ideal structure of a tensor product of certain algebras
I would be grateful if anyone could give me a reference regarding the following question.
Suppose that $A$ and $B$ are two unital prime algebras over a field $F$ whose center consists of scalar ...
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Ideal generated by two univariate, coprime, integer polynomials
Let $f(x)$, $g(x)$ be two univariate, coprime, integer polynomials and let $I=\big(f(x),g(x)\big)$ the ideal of $\mathbb{Z}[x]$ generated by $f, g$. Let $I \cap \mathbb{Z}$, that is, the elements of $\...
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A property of minimal prime ideals in commutative reduced ring
Let $\{\mathfrak{p}_i\}_{i\in I}$ ($I$ is an infinite set) be a family of minimal prime ideals in a commutative reduced ring $R$ with identity, and let $a, b \in R$. If the ideal $\langle a, b\rangle$...
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a question on symbolic power
Let $R$ be a Noetherian ring and $P$ a prime ideal. Then the $n$-th symbolic power of $P$ is
$$P^{(n)} = P^n R_P \cap R = \{ f \mid sf \in P^n \text{ for some } s \in R - P\}$$
(cf. wiki). We have $P^...
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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 ...
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Adjunctions capturing duality between ideals and saturated monoids in a commutative ring?
Let $R$ be a commutative ring. A saturated monoid in $R$ is a multiplicative submonoid $S\subset R$ which is closed under divisors, i.e $xy\in S\implies x\in S$. This is the converse of the analogous ...
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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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When $D$ and $D[x]$ have isomorphic $t$-class groups
In a 1987 paper, Stefania Gabelli studies the map
$$\psi: \{ \text{ideals of $D$} \} \rightarrow\{\text{ideals of $D[x]$\}}$$
$$\psi(I) = ID[x]$$
for a domain $D$. In fact $\psi$ always induces an ...
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Can the Artin-Rees lemma be derived from Krull Intersection theorem?
The Krull Intersection theorem states that : For a finitely generated module $M$ over a Noetherian ring $R$ and any ideal $I$ of $R$, we have $I(\cap_{k=1}^{\infty}I^k M)=\cap_{k=1}^{\infty}I^k M$ . ...
user111524
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Normal ideals on $[\lambda]^{<\kappa}$ concentrating on maximal cardinality
Is the following consistent? $2^\omega > \omega_2$, and there is a normal precipitous ideal $I$ on $[\omega_2]^\omega$ such that every $X \subset [\omega_2]^\omega$ of size $< 2^\omega$ is in $...
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Proof in Schertz's Complex Multiplication
I'm reading through Complex Multiplication by Reinhard Schertz, and I'm stuck at Theorem 3.1.8.
Let $\mathfrak{O}_t$ be the order of conductor $t$ in an imaginary quadratic field $K$.
He defines ...
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$I,J$ are $p$-primary ideals, but $I+J$ is not
I asked this question on the stack exchange, and after no answers and the recommendation of someone else, I am posting it here on MO. I am looking for an example of two ideals $I$ and $J$ in a ...
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Some questions about homogroups
Every semigroup containing an ideal subgroup is called a homogroup. Let $(S,\cdot)$ be homomgroup, hence it contains an ideal $I$ that is also a subgroup. It is easy to see that $I$ is the least ideal,...
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Is there a C*-algebra whose Pedersen ideal is not proper?
In [1, 5.6.3] Pedersen states without proof or reference that there are non-unital C*-algebras whose Pedersen
ideal is the whole algebra.
Does anyone know where can I find such an example?
Is it ...
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Condition for a monomial to belong to a particular ideal
Consider the polynomial ring $R[x_1,x_2,\ldots,x_n]$, where $R$ be an algebraically closed field (preferably $\mathbb{C}$) and the ideal $J=\langle m_1, m_2,\ldots,m_n\rangle$ generated by monomials . ...
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Finite distributive lattices as lattice of ideals of a finite ring
Is there a finite distributive lattice that is not isomorphic to the lattice of ideals of a finite ring?
- 45.4k
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Can a minimal generating set for an ideal always be made into a Groebner basis?
Let $I\subseteq k[x_0,\ldots,x_n]$ be an ideal, generated by some polynomials $F_1,\ldots,F_r$, all homogeneous and of the same degree. Suppose $r$ is the smallest number of generators that will ...
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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 ...
user21167
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Noetherian ring with a "strange" idempotent ideal
Do you know a left-noetherian ring $R$ with a two-sided ideal $I$ such that:
$I=I.I$;
$I$ is not projective as a left $R$-module (and better, the tensor product over $R$ of $I$ with itself is not a ...
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How can we find a monic polynomial with the smallest degree in left ideal of $\mathrm{Mat}(F[x])$?
Let $F$ be a finite field, $R=F[x]$ be a polynomial ring and $K = \mathrm{Mat}_n(R)$ be a full matrix ring over $R$. We identify the ring $K$ with the ring $\mathrm{Mat}_n(F)[x]$, for example
$$
\left(...
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When $\langle u,v,w \rangle$ is a maximal ideal in $\mathbb{C}[x,y]$?
Let $u,v,w \in \mathbb{C}[x,y]$ and let $\langle u,v,w \rangle$ be the ideal generated by $u,v,w$.
It is known that for two elements the following result holds:
$\langle u,v \rangle$ is a maximal ...
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Which projections maintain irreducibility of the polynomial $x_0x_1 + x_2x_3 + \dots + x_{2m-2}x_{2m-1}$?
Let $q = x_0x_1 + x_2x_3 + \dots + x_{2m-2}x_{2m-1} \in \Bbb{C}[x_0, \dots, x_{2m-1}]$. The ambient space is $\Bbb{C}^{2m}$. Which are the polynomials $p \in \Bbb{C}[x_0, \dots, x_{2m-1}]$ such that ...
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Saturated ideals in computational algebra
Let $R$ a commutative ring with one and $I, J \triangleleft R$ two ideals.
The saturated ideal $I^{sat}_J$ with respect $J$ is the ideal
$$(I : J^\infty )= \cup_{n \geq 1} (I:J^n)$$
where $(I:J^n)= \{...
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Noetherian rings as homomorphic image of finite direct product of Noetherian domains?
A theorem of Hungerford says that : Every PIR (principal ideal ring , obviously commutative ) is a homomorphic image of a finite direct product of PID s . My question is , is there a similar criteria ...
user111524
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$T$-nilpotent ideals
Recall that a subset $I$ of a ring $R$ is left (resp.,
right) $T$-nilpotent in case for every sequence $$a_1,a_2,\cdots $$
in $I$ there is an $n$ such that $a_1\cdots a_n=0$ (resp.,
$a_n\cdots a_1=0$)....
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commutative, infinite, artinian ring (with unity) in which distinct ideals has distinct index
Let $R$ be an infinite commutative Artinian ring such that for any two distinct ideals $I, J$ of $R$, $R/I$ and $R/J$ has different cardinalities; then is it true that $R$ is a PIR (principal ideal ...
user111524
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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 ...
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Why does the variety of ideals in this quaternion type algebra have a non-reduced structure?
Let $A$ be the $\mathbb{C}$-algebra generated by elements $i,j$ with relations $i^2=j^2=0$ and $ij=-ji$, i.e. we have $A=\mathbb{C}\oplus\mathbb{C}i\oplus\mathbb{C}j\oplus\mathbb{C}ij$.
Let $\mathcal{...
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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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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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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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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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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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