# Questions tagged [ideals]

The ideals tag has no usage guidance.

**3**

votes

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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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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)$A(SO(3))$