# Questions tagged [ideals]

The tag has no usage guidance.

114 questions
Filter by
Sorted by
Tagged with
762 views

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 ...
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 ...
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,...
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 ...
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 ... 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 ...
1 vote
87 views

72 views

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 $b\not\in I^... 1 vote 0 answers 66 views ### When do the kernels of module homomorphisms between rings whose kernels contain a given fixed ideal contain every prime ideal over it?$\DeclareMathOperator{\Hom}{Hom}$All our rings are commutative with unity and, if necessary, we can suppose that they are actually polynomial rings over a field in finitely many variables where the ... 3 votes 1 answer 164 views ### Is the kernel of an action of a Hopf algebra on an algebra a biideal? I think, this must be simple, but I am not a specialist in this field, so excuse me. I asked this a week ago at MSE, but without success. S.Dascalescu, C.Nastasescu and S.Raianu define the action of a ... 4 votes 1 answer 341 views ### On the annihilator of a module Question. Let$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)$? Remark. The annihilator of a ... 1 vote 0 answers 128 views ### Concatenation of two radical ideals is radical Let$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$... 4 votes 1 answer 171 views ### Is there a C*-algebra whose Pedersen ideal is not proper? In [1, 5.6.3] Pedersen states without proof or reference that there are non-unital C*-algebras whose Pedersen ideal is the whole algebra. Does anyone know where can I find such an example? Is it ... 3 votes 1 answer 307 views ### Prime ideals and localizations of the ring$\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$-th prime. This is how far I got: I could conclude, by means of the ... 3 votes 0 answers 124 views ### Quick proof of the first part of Kaplansky's Theorem on characterization of Noetherian domains with all maximal ideals principal I have been reading section 12 of this paper "Elementary Divisors and Modules" by I. Kaplansky (https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-... 2 votes 1 answer 350 views ### A question on a Macaulay2 computation I have an ideal$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$\... 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 ...
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?
1 vote
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 ...
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 ...
101 views

467 views

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 ...
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 ...
1 vote
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....
85 views

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 ...
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 . ...
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 ...
1 vote
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 $e=ry$. ...
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 ...
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 ...
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 ...
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 ...
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 $x\in S$; ...
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 ...