Questions tagged [ideals]

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9 votes
2 answers
762 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 $\...
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2 votes
1 answer
76 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'm also interested in a more general form of this statement. Let $A$ be a commutative algebra (over $\mathbb{R}$ or $\mathbb{C}$), $I$ is ...
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5 votes
0 answers
111 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 ...
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0 votes
1 answer
86 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 \;\&\; ...
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  • 617
7 votes
1 answer
239 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 ...
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  • 10.9k
2 votes
1 answer
166 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 ...
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  • 127
3 votes
0 answers
110 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,...
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2 votes
1 answer
123 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 ...
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4 votes
1 answer
193 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 ...
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2 votes
1 answer
129 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 ...
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1 vote
1 answer
87 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)}=(...
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  • 1,643
8 votes
2 answers
735 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 ...
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1 vote
1 answer
193 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 ...
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  • 113
2 votes
1 answer
141 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_{...
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  • 287
0 votes
0 answers
72 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^...
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  • 1,535
1 vote
0 answers
66 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 ...
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  • 522
3 votes
1 answer
164 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 ...
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4 votes
1 answer
341 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 ...
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  • 183
1 vote
0 answers
128 views

Concatenation of two radical ideals is radical

Let $I = (f_1,f_2,\cdots, f_k) \subseteq \mathbb{C}[x_1,x_2,\cdots, x_m]$ and $J = (g_1,g_2,\cdots, g_r) \subseteq \mathbb{C}[y_1,y_2,\cdots, y_n]$ be radical ideals (we know that the $f_i$ and $g_j$ ...
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  • 131
4 votes
1 answer
171 views

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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  • 441
3 votes
1 answer
307 views

Prime ideals and localizations of the ring $\mathbb Z[\{\sqrt p: p \text{ prime}\}]$

I have been trying to study the prime ideals of the ring $R:=\mathbb Z [\{ \sqrt{p_n}\}_{n=1}^\infty]$, where $p_n$ denotes the $n$-th prime. This is how far I got: I could conclude, by means of the ...
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3 votes
0 answers
124 views

Quick proof of the first part of Kaplansky's Theorem on characterization of Noetherian domains with all maximal ideals principal

I have been reading section 12 of this paper "Elementary Divisors and Modules" by I. Kaplansky (https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-...
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2 votes
1 answer
350 views

A question on a Macaulay2 computation

I have an ideal $I$ generated by quadratic and cubic homogeneous polynomials in $10$ variables. Macaulay2 tells me that $I$ defines an irreducible variety $X$ of dimension $5$ and degree $10$ in $\...
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4 votes
1 answer
436 views

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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  • 441
2 votes
0 answers
65 views

Properties preserved in addition of ideals

If I and J are prime (radical) ideals then what are the conditions under which we can define a prime (radical) ideal from I+J?
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1 vote
0 answers
189 views

Ideals of maximal tensor product of $C^{\ast}$-algebras

Let $A$ and $B$ be $C^{\ast}$-algebras. It is well known that maximal tensor product of simple $C^{\ast}$-algebras need not be simple. So basically the ideal structure of $A\otimes_{max}B$ does not ...
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2 votes
1 answer
161 views

For an integral domain $R$ when does $a^2 \equiv b^2 \bmod 4R$ imply $a \equiv b \bmod 2R$?

Suppose we have $a^2 \equiv b^2 \bmod 4R$ where $R$ is an integral domain. Under what conditions on $R$ can we conclude that $a \equiv b \bmod 2R$? This would hold if $2 \in R$ is a prime or the ...
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  • 97
3 votes
0 answers
101 views

Is the determinantal ideal of the span of a linearly independent set of rank-one matrices radical?

Let $k$ be an algebraically closed field, and let $X_1,\dots, X_n \in M_m(k)$ be rank-one matrices that are linearly independent over $k$. For a fixed integer $1 \leq r \leq m$, consider the ideal $I \...
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  • 874
3 votes
1 answer
91 views

Modular and primitive ideals of $C_{0}(X,A)$

Let $A$ be $C^{\ast}$- Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\...
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3 votes
0 answers
467 views

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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2 votes
2 answers
139 views

Results which are known about ideals of spatial tensor product

I am studying about ideals of spatial (minimal) tensor product of $C^{\ast}$-algebras but I did not find any book/paper in which all the results are given. What are some results or folklore which ...
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0 votes
0 answers
67 views

Need reference of books which deals with ideal theory of tensor product

Is there any book which deals with Ideal theory of tensor product of $C^{\ast}-$ algebras
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2 votes
1 answer
125 views

Norm of a multiplier of a right-ideal in C*-algebras

Let $A$ be a $C^*$-algebra. If $I$ is an essential two-sided ideal in $A$, then it is fact that for every $a \in A$ we have $\|a\| = \sup_{x \in I, \|x\|=1} \|xa\|$. The argument is that we have an ...
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  • 2,646
0 votes
1 answer
161 views

Need reference for ideals and representations of $C_0(X,A)$

Let $A$ be $C^{\ast}$- Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\...
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2 votes
1 answer
117 views

Principal ideal of a non-associative magma

The definitions of an ideal of an algebraic structure $A$ (as a substructure $I$ such that the product of $A$ and $I$ is a subset of $I$) do not involve associativity. However, the definitions of a ...
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  • 133
5 votes
1 answer
136 views

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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  • 9,655
1 vote
0 answers
101 views

Relation between Betti Numbers and Chromatic Number of a simple graph

Is there a relation between the betti numbers of a graph considered as a simplicial complex and its chromatic number? Typically the first Betti number is said to be the cyclomatic number of the graph....
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  • 1,611
0 votes
0 answers
85 views

An ideal invariant under an automorphism

The following question appears here; hopefully, it is appropriate for MO. Let $k$ be a field of characteristic zero, and let $\beta: k[x,y] \to k[x,y]$ be the following involution $\beta: (x,y) \...
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  • 2,573
4 votes
2 answers
274 views

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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  • 811
2 votes
1 answer
98 views

Ramsey type properties of $F_\sigma$ ideals

Let $I \subseteq 2^\omega$ be any $F_\sigma$ ideal containing every finite sets : $\forall X \in I\ \forall Y \subseteq X\ \text{ we have } Y \in I$ $\forall k\ \forall X_1,\dots,X_k \in I\ X_1 \cup \...
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2 votes
0 answers
88 views

Invertibility modulo the intersection of ideals in $C^*$-algebras

This is a crosspost from math.se because it did not get any attention whatsoever. I therefore assume that it fits here better. Let $\mathcal{A}$ be a $C^*$-algebra and $A \in \mathcal{A}$. I am ...
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  • 171
4 votes
1 answer
162 views

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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  • 1,611
0 votes
1 answer
175 views

Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?

Note: Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a (very ...
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1 vote
1 answer
159 views

Special idempotents in a commutative ring

Let $R$ be a commutative ring with $1$ and $e$ be an idempotent element of $R$ with the property that if $e=x+y$ (where $x, y\in R$), then there exists $r\in R$ such that either $e=rx$ or $e=ry$. ...
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4 votes
1 answer
231 views

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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4 votes
1 answer
199 views

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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  • 153
5 votes
3 answers
1k views

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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  • 383
2 votes
0 answers
159 views

Does a surjection between polynomial rings lose one Krull dimension per generator quotiented?

Let $k$ be a Noetherian commutative ring with unity (I am happy adding hypotheses, as I will ultimately want $k = \mathbb Z$) and $$\varphi\colon A = k[x_1,\ldots,x_n] \to k[y_1,\ldots,y_p] = B$$ a ...
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  • 2,939
2 votes
1 answer
82 views

Is the Scott topology generated by the ideals as the closed sets?

Let $X$ be a directed-complete partial order, or even a complete lattice. A subset $S\subseteq X$ is called Scott-closed if and only if it is: Downward-closed: $y\in S$ and $x\le y$ implies $x\in S$; ...
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  • 1,999
13 votes
1 answer
445 views

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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