Questions tagged [ideals]

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51 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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47 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 ...
3
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1answer
134 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 ...
4
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1answer
237 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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0answers
124 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 $g_j$ ...
4
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1answer
119 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 ...
2
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1answer
173 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 ...
3
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0answers
70 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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1answer
239 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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43 views

Maximal left ideals of free associative algebras

What are the maximal left ideals of a free associative, unital algebra over a field?
4
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1answer
391 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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64 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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119 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 ...
2
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1answer
160 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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90 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 \...
3
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1answer
80 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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165 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)= \{...
2
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2answers
108 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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58 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
2
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1answer
117 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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1answer
138 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)}^{\...
2
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1answer
79 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 ...
5
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1answer
123 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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0answers
92 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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80 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) \...
4
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2answers
245 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 ...
2
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1answer
91 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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83 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 ...
4
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1answer
160 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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1answer
154 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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1answer
155 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 $e=ry$. ...
4
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1answer
206 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 ...
4
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1answer
138 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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3answers
656 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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0answers
138 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 ...
2
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1answer
63 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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1answer
390 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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0answers
90 views

Similar reduced integral matrices

Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example), but if necessary I'm willing to reduce to the case where $d$ is squarefree and $-...
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2answers
352 views

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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0answers
56 views

Efficient algorithm to prove that a polynomial ideal contains 1

I have the following problem: Suppose to have an ideal $I\triangleleft k[x_1,...,x_n]$ defined by generators. There exists an efficient algorithm (perhaps more efficient than calculating the Groebner ...
7
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128 views

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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0answers
135 views

On finite dimensional commutative algebras and regular sequences

Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...
2
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1answer
169 views

Homomorphisms of ring extending nicely ideal intersections

Let $\varphi\!:\!S\to R$ be a homomorphisms of $K$-algebras for some field $K$. Let $\{a_{\lambda}\}_{\lambda}$ be a family of ideals of $S$. Is there some "natural" assumption on $\varphi$ to ...
0
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1answer
107 views

left ideals in Lie super algebras

Let $\mathfrak g$ be a Lie superalgebra. If $\mathfrak a$ is not a grade subspace of $\mathfrak g$, then why does $[\mathfrak g, \mathfrak a]$ and $[\mathfrak a, \mathfrak g]$ are not same? For me ...
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1answer
97 views

Finding a characteristic for which the zero-locus of an ideal is not empty

I have a set of polynomials $f_1, \dots, f_m \in \mathbb{Z}[x_1, \dots, x_n]$ and I am interested in finding if these polynomials have a common root inside either $\mathbb{C}[x_1, \dots, x_n]$ or $\...
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0answers
169 views

Ideals and Idempotents in a commutative ring

Let $I$, $J$, and $K$ be pairwise comaximal ideals of a commutative ring $R$ with $1$ with the property that if $x^2-x\in I$, then there exists $e^2=e\in J$ such that $x-e\in I$ and if $y^2-y\in K$, ...
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1answer
164 views

Is this algorithm for primary decomposition correct?

I've written some code for Sage to compute radical ideals and primary decompositions over $\overline{Q}$ (the field of algebraic numbers), and I'm not sure if it's right. Since Singular (the ...
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2answers
553 views

Reduced ring with all non-prime ideals finitely generated

Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ? Without reduced assumption, it is not true even ...
3
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1answer
152 views

Solving system of multivariable algebraic equations over $\mathbb Q$ by reducing over $\mathbb F_p$

I try to solve the finite system of multivariable algebraic equations with coefficients from $\mathbb Q$. It would be sufficient for me to prove that there is only finite number of solutions over $\...
2
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0answers
281 views

Intersection of all positive powers of prime ideal in an integral domain with all ideals of finite height

Let $R$ be an integral domain with every prime ideal having finite height. Then is $\bigcap_{n>1} P^n$ a prime ideal of $R$ for every prime ideal $P$ of $R$ ? If that is not true in general, then ...