Questions tagged [ideals]

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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 ...
Fred.Fred's user avatar
  • 409
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356 views

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 ...
Lisa's user avatar
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5 votes
0 answers
130 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 ...
Math Lover's user avatar
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60 views

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 ...
Badam Baplan's user avatar
5 votes
0 answers
325 views

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 $I(\cap_{k=1}^{\infty}I^k M)=\cap_{k=1}^{\infty}I^k M$ . ...
user avatar
5 votes
0 answers
142 views

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 $...
Monroe Eskew's user avatar
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4 votes
0 answers
220 views

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 ...
user237522's user avatar
  • 2,783
4 votes
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119 views

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 ...
Varun Ramanathan's user avatar
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1k 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)= \{...
user267839's user avatar
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276 views

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 ...
user avatar
4 votes
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472 views

$T$-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., $a_n\cdots a_1=0$)....
Sebastian Burciu's user avatar
3 votes
0 answers
91 views

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 $...
Salvo Tringali's user avatar
3 votes
0 answers
202 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-...
asrxiiviii's user avatar
3 votes
0 answers
111 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 \...
Ben's user avatar
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3 votes
0 answers
108 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 $-...
GreginGre's user avatar
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3 votes
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186 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 $y^2-y\in K$, ...
E.R's user avatar
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0 answers
79 views

Semigroups containing an ideal with a local identity

I'm looking for some classes of semigroups containing a (proper) ideal with a local identity (i.e., ideal submonoid). Can somebody give some examples or/and theorems for the followings cases: (a) ...
M.H.Hooshmand's user avatar
2 votes
0 answers
69 views

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 $g(x_1,\ldots, x_m),\, g(x_2,\ldots, x_{m+1}), \ldots, g(x_{1+{n-m}},\ldots , x_{n})$? ...
olha's user avatar
  • 21
2 votes
0 answers
121 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,...
Sylvain JULIEN's user avatar
2 votes
0 answers
143 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 $g_j$ ...
atenao's user avatar
  • 173
2 votes
0 answers
67 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?
Iqra Khan's user avatar
2 votes
0 answers
278 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 ...
Math Lover's user avatar
  • 1,065
2 votes
0 answers
94 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 ...
Klaus's user avatar
  • 171
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0 answers
179 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 ...
jdc's user avatar
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2 votes
0 answers
61 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 ...
DDT's user avatar
  • 297
2 votes
0 answers
393 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 ...
user avatar
2 votes
0 answers
75 views

On GCD and LCM of elements in integral domains which has the property that any over ring has Going Down

Let $R$ be an integral domain with fraction field $K$ such that for every ring $R \subseteq S \subseteq K$ , the ring extension $R \subseteq S$ satisfies Going Down property i.e. for any chain of ...
user avatar
2 votes
0 answers
103 views

Non-atomicity of submeasures associated to analytic P-ideals

Let $\phi$ be a lower semicontinuous submeasure, that is, a function $\mathcal{P}(\mathbf{N}) \to [0,\infty]$ which is monotone, subadditive, and $$ \phi(A)=\lim_{n\to \infty} \phi(A \cap [1,n]) $$ ...
Paolo Leonetti's user avatar
2 votes
0 answers
173 views

A property of families of finitely generated ideals

For a commutative unital ring $R$, let $J(R)=0$ (a semiprimitive ring), and for any family of finitely generated ideals $\{I_i\}$ if $\cap I_i=0$, then a finite intersection of $\{I_i\}$'s is also ...
Andro Zimone's user avatar
2 votes
0 answers
75 views

How are Polynomials of Toric ideals Studied with Exponents as ST-cuts?

Topic: Toric ideals on Expected value of Structure Functions in Random Graphs Goal: to understand the toric ideal where the exponents $h_i$ and $s_j$ are st-vertex-cuts of a digraph \begin{equation} ...
hhh's user avatar
  • 143
2 votes
0 answers
131 views

some sort of 'saturation' of module quotients

Let $R$ be a local Noetherian ring over a field, with the maximal ideal $\mathfrak{m}$. (e.g. $R=k[[x_1,\dots,x_{p>1}]]$) Given two $R$-modules, $N\subset M$, of the same (finite, non-zero) rank. ...
Dmitry Kerner's user avatar
1 vote
0 answers
52 views
+50

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 $R$ ...
Alex's user avatar
  • 323
1 vote
0 answers
50 views

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 $x^{-1}$...
Ralle's user avatar
  • 471
1 vote
0 answers
122 views

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 (...
Zhang Yuhan's user avatar
1 vote
0 answers
212 views

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 $k\in\mathbb{N}$ ...
Sanae Kochiya's user avatar
1 vote
0 answers
63 views

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 ...
Chess's user avatar
  • 13
1 vote
0 answers
38 views

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 ...
George Bentley's user avatar
1 vote
0 answers
92 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 ...
Hvjurthuk's user avatar
  • 573
1 vote
0 answers
121 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....
vidyarthi's user avatar
  • 2,007
1 vote
0 answers
138 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 ...
BrianT's user avatar
  • 1,197
1 vote
0 answers
135 views

Monomorphism between two ideals

Let $I $ and $J $ be two ideals in a commutative ring $R $ with $1$. Is there any equivalent property for the fact that there are no $R $-module monomorphism from $I $ to $J $?
Artor's user avatar
  • 21
1 vote
0 answers
315 views

Find the generators of a complete intersection maximal ideal

Let $k$ be a field (not necessarily algebraically closed), define the $k$-algebra $$ B:=k[x_{0}, x_{1}, x_{2}, x_{3}]_{(F_{2}F_{3})} $$ (degree 0 part of the localization), it's the coordinate ring of ...
Lao-tzu's user avatar
  • 1,856
1 vote
0 answers
91 views

Cut ideal of two graphs?

Consider a connected graph $G$ and a connected graph $H$. Their graph ideals are their path ideals. The Alexander duality of the graph ideals give the cut ideals. $G$ and $H$ are not connected to each ...
hhh's user avatar
  • 143
0 votes
0 answers
56 views

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? ...
PPenguin's user avatar
  • 101
0 votes
0 answers
32 views

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 $gcd(N^2, nrd(α)) = N$....
student17's user avatar
0 votes
0 answers
82 views

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]/...
Sergiy Maksymenko's user avatar
0 votes
0 answers
145 views

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 ...
Ethan's user avatar
  • 1
0 votes
0 answers
74 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^...
Fallen Apart's user avatar
  • 1,605
0 votes
0 answers
74 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
Math Lover's user avatar
  • 1,065
0 votes
0 answers
102 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) \...
user237522's user avatar
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