Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [ideals]

The tag has no usage guidance.

3
votes
0answers
60 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) For all $a,b,c\in\mathbb{Z}$ such that $ac-b^2=d,$ set $[a,b,c]_d=\begin{pmatrix}...
4
votes
2answers
217 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}...
2
votes
0answers
46 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 ...
8
votes
0answers
119 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 ...
2
votes
0answers
116 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
votes
1answer
107 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
votes
1answer
84 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 ...
1
vote
1answer
78 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 $\...
3
votes
0answers
143 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 $y^2-y\in K$, ...
1
vote
1answer
91 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 ...
-2
votes
2answers
349 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
votes
1answer
142 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
votes
0answers
118 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 ...
0
votes
0answers
36 views

On the relation of ideals and $\mathcal J$-classes in semigroups

Given a semigroup $S$, a subset $I$ is called an ideal iff for every $s \in S$ we have $sI, Is \subseteq I$. Further we set $$ s \le_{\mathcal J} t :\Leftrightarrow SsS \cup \{s\} \subseteq StS \cup \...
1
vote
1answer
137 views

Valuation ring whose maximal ideal and every ideal of finite height are principal

Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
2
votes
1answer
129 views

On minimal generating sets of certain submodules

All our rings are commutative with unity. For an $R$-module $M$ and a submodule $N$ of $M$ and ideal $I$ of $R$, let $(N:I):=\{m\in M : Im \subseteq N\}$. Let $\mu (M)$ denote the least cardinality ...
3
votes
0answers
31 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 ...
0
votes
0answers
57 views

On the set of radical ideals which is the radical of an ideal generated by atleast a fixed cardinal no. many generators

Let $R$ be a commutative ring with unity and $\alpha$ be an infinite cardinal. For any ideal $J$ of $R$, let $\mu(J)$ denote the minimal cardinality among generating sets of $J$. For $I$ an ideal of $...
0
votes
0answers
21 views

On certain ideals generated by two elements in Prufer domain

Let $R$ be a Prufer domain. Let $0\ne a,b \in R$ be such that for every $c \in R$, either $abR+cR=R$ or else $abR+cR$ is contained in a proper principal ideal. Now suppose $c \in R$ be such that $aR+...
5
votes
0answers
184 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 $I(\cap_{k=1}^{\infty}I^k M)=\cap_{k=1}^{\infty}I^k M$ . ...
2
votes
3answers
218 views

GCD and LCM of elements in Prufer domain

Let $R$ be a Prufer domain. If $0 \ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is also principal for every $b\in R$ ? Over Prufer ...
2
votes
0answers
51 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 ...
6
votes
1answer
234 views

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 $...
4
votes
1answer
212 views

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?
2
votes
1answer
147 views

A relation between annihilators and ideals

Let $I$ and $J$ be two ideal of a ring $R$ (commutative with $1$) such that $I\subseteq Ann_R(Ann_R(J))$ and $I$ is a principal ideal. Is there any conditions on $I$ or $J$ or both of them under which ...
6
votes
1answer
375 views

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 ...
2
votes
1answer
127 views

Commutative rings with unity over which every non-zero module has an associated prime

Let $R$ be a commutative ring with unity such that every non-zero module over $R$ has an associated prime. Then is it true that $R$ is Noetherian ? If not, then can we say something about any possible ...
3
votes
1answer
113 views

Indecomposable quotient of Prüfer domains

Let $D$ be a Prüfer domain. I am looking for equivalent condition on an ideal $I$ of $D $ under which $D/I $ is an indecomposable ring.
4
votes
1answer
142 views

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,...
6
votes
0answers
171 views

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 ...
3
votes
1answer
180 views

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 ...
3
votes
0answers
131 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 ...
2
votes
1answer
218 views

On graded projective modules

If $R$ is a PID, then we know that any projective module over $R$ is free. Is there any graded version of this? That is, let us call a graded commutative ring $R=\oplus _{g \in G} R_g$ a graded PID if ...
2
votes
1answer
106 views

On maximal ideals of $k [X_i : i \in I ] $ where $k$ is a field , $I$ is an infinite set with $|k| > |I|$

Let $k$ be a field and $I$ be an infinite set such that $|k| > |I|$ . Let $R := k [X_i : i \in I ] $ and $m$ be a maximal ideal of $R$ ; then is it true that $m \cap k[X_i] \ne 0 , \forall i \in I$ ...
2
votes
1answer
203 views

On elements of a domain which satisfy a condition of Kummer

Let $R$ be an integral domain. Let $k(R)$ be the set of elements $a \in R \setminus \{0\}$ such that for every $b \in R$, either $Ra + Rb = R$ or $Ra + Rb$ is contained in a proper principal ideal. ...
3
votes
0answers
66 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) ...
13
votes
2answers
345 views

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 ...
4
votes
1answer
156 views

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(...
1
vote
1answer
130 views

On multiplicative closedness of a special set of elements in integral domains

Let $D$ be a domain which is not a field. Let $i(D):=\{a\in D \setminus \{0\} :$ for every ideal $I$ of $D$ containing $a$, there exist $b \in I$ such that $I=Da+Db$ $ \}$. My question is: Is $i(...
2
votes
1answer
440 views

On the set of non-zero elements in an integral domain whose generating principal ideal is of a special kind

Let $R$ be an integral domain. Consider the set $$S := \big\{a \in R\smallsetminus \{0\} : Ra+Rx \text{ is a principal ideal } \forall x \in R \big \}.$$ Is $S$ a saturated multiplicative closed ...
2
votes
0answers
99 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 $$ \phi(A)=\lim_{n\to \infty} \phi(A \cap [1,n]) $$ ...
3
votes
1answer
247 views

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 ...
5
votes
0answers
133 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 $...
1
vote
0answers
91 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 $?
13
votes
1answer
454 views

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 $n$...
1
vote
0answers
188 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 ...
1
vote
1answer
87 views

Dimension of quotient of ideals

Let $A$ be a Banach algebra, $I$ be a closed two-sided ideal in $A$, and $J$ be a closed two-sided ideal in $I$ such that there is no ideal between $I$ and $J$. Can we see $dim(\frac{I}{J})<\...
2
votes
1answer
116 views

Transitivity of ideals of banach algebra

Let $A$ be a Banach algebra, $I$ be a closed two-sided ideal in $A$ and $J$ be a closed two-sided ideal in $I$. When $J$ is a closed two-sided ideal in $A$? (Except modes $I$ be a complemented ideal ...
4
votes
1answer
240 views

$I,J$ 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 ...
1
vote
1answer
107 views

Ideals of Banach algebras

Is there any characterization of ideals space or maximal ideals space of following Banach algebras? 1)$C^1{([0,1])}$ 2) disc algebra 3)$A(SO(3))$