All Questions
Tagged with ac.commutative-algebra ideals
86 questions
21
votes
4
answers
5k
views
The number of ideals in a ring
Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here.
Let $R$ be a finite commutative ring with identity. Under what conditions the number of ...
- 1,881
13
votes
1
answer
595
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 ...
- 943
13
votes
1
answer
940
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 $n$...
- 1,802
10
votes
2
answers
1k
views
Do relatively prime polynomials $f$ and $g$ in $k[x,y]$ generate an ideal of finite codimension?
Let $k$ be a field and $f,g \in k[x,y]$ be two non-constant polynomials in two variables. Is it true that the following conditions are equivalent:
$f$ and $g$ are relatively prime in $k[x,y]$, in the ...
7
votes
2
answers
450
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}...
- 11.1k
7
votes
1
answer
170
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 ...
- 211
7
votes
0
answers
275
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 ...
- 409
7
votes
0
answers
369
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 ...
- 71
6
votes
1
answer
672
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 ...
- 69
6
votes
1
answer
288
views
What is the right level of generality for $(R/a) \times (R/b) \cong (R/\gcd(a,b)) \times (R/\operatorname{lcm}(a,b))$?
Let $R$ be a principal ideal domain. Let $a$ and $b$ be two elements of $R$. Let $g$ be a greatest common divisor of $a$ and $b$, and let $\ell$ be a least common multiple of $a$ and $b$. (Of course, ...
- 33.8k
6
votes
1
answer
355
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 $...
user111524
5
votes
1
answer
1k
views
Ideal generated by two univariate, coprime, integer polynomials
Let $f(x)$, $g(x)$ be two univariate, coprime, integer polynomials and let $I=\big(f(x),g(x)\big)$ the ideal of $\mathbb{Z}[x]$ generated by $f, g$. Let $I \cap \mathbb{Z}$, that is, the elements of $\...
- 7,746
5
votes
1
answer
1k
views
A property of minimal prime ideals in commutative reduced ring
Let $\{\mathfrak{p}_i\}_{i\in I}$ ($I$ is an infinite set) be a family of minimal prime ideals in a commutative reduced ring $R$ with identity, and let $a, b \in R$. If the ideal $\langle a, b\rangle$...
- 51
5
votes
1
answer
291
views
a question on symbolic power
Let $R$ be a Noetherian ring and $P$ a prime ideal. Then the $n$-th symbolic power of $P$ is
$$P^{(n)} = P^n R_P \cap R = \{ f \mid sf \in P^n \text{ for some } s \in R - P\}$$
(cf. wiki). We have $P^...
- 2,773
5
votes
1
answer
759
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 ...
- 303
5
votes
1
answer
190
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 ...
- 10.5k
5
votes
0
answers
2k
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)= \{...
- 6,028
5
votes
0
answers
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 ...
- 938
5
votes
0
answers
337
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 $I(\cap_{k=1}^{\infty}I^k M)=\cap_{k=1}^{\infty}I^k M$ . ...
user111524
4
votes
2
answers
310
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 ...
- 901
4
votes
1
answer
677
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 ...
- 61
4
votes
1
answer
256
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 . ...
- 2,089
4
votes
0
answers
140
views
Can an ideal in the ring of holomorphic functions on the complex plane be non-finitely generated?
Let $( I )$ be an ideal in the ring $( R )$ of all holomorphic functions of a single complex variable on the complex plane. I am interested in understanding whether it is possible for $( I )$ to be ...
- 93
4
votes
0
answers
238
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 ...
- 2,837
4
votes
0
answers
280
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 ...
user111524
3
votes
1
answer
339
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 ...
user111524
3
votes
1
answer
227
views
Van der Waerden's classical paper on Hilbert polynomials and Bezout's theorem
Is it possible to obtain an electronic copy of Van der Waerden's "On Hilbert series of composition of ideals and generalisation of the Theorem of Bezout", Proceedings of the Koninklijke ...
- 5,339
3
votes
1
answer
213
views
Ideals whose quotient rings have a certain property
There are some well-known properties of ideals which are equally well-known to correspond to properties of their respective quotient rings. For example:
An ideal $p$ of a ring $R$ is prime iff $R/p$ ...
- 559
3
votes
1
answer
257
views
Does the set of ideals, whose Jacobson radical & nilradicals coincide, form a sublattice?
This question might be below the level of MO, so apologies in advance. I posted the same question in MS about a week ago without an answer so far.
Let $R$ be a unital commutative ring and $L(R)$ ...
- 2,605
3
votes
1
answer
394
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 ...
- 165
3
votes
1
answer
606
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 ...
- 739
3
votes
1
answer
128
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.
- 61
3
votes
1
answer
701
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 ...
user111524
3
votes
0
answers
245
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-...
- 739
3
votes
0
answers
115
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 \...
- 980
3
votes
0
answers
188
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 $y^2-y\in K$, ...
- 31
2
votes
3
answers
304
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 ...
user111524
2
votes
1
answer
168
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 ...
- 97
2
votes
1
answer
138
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 $m \cap k[X_i] \ne 0 , \forall i \in I$ ...
user111492
2
votes
1
answer
166
views
DCC on the powers of ideals
My apologies if this question is below the level of MO. I posted the same question in MS about a week ago without an answer so far.
Let $R$ be a unital commutative ring. $R$ is called strongly $\pi$-...
- 2,605
2
votes
1
answer
150
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 ...
user111524
2
votes
1
answer
232
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 ...
user111492
2
votes
1
answer
279
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 ...
- 33.8k
2
votes
1
answer
216
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 ...
- 43
2
votes
1
answer
225
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.
A ...
user111524
2
votes
1
answer
2k
views
In what conditions every ideal is an extension ideal? Is every prime ideal extension of prime ideal?
Let $R$ and $S$ be commutative rings (with $1$), and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). When $f$ is ...
- 1,355
2
votes
1
answer
323
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 ...
- 43
2
votes
0
answers
70
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?
- 51
2
votes
0
answers
183
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 ...
- 2,995
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 ...
- 297