All Questions
Tagged with ac.commutative-algebra ideals
86 questions
1
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394
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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 ...
- 415
-2
votes
2
answers
763
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 ...
user111492
2
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0
answers
407
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 ...
user111492
1
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1
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463
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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 ?
user111492
2
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1
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150
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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
5
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0
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60
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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
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0
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337
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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
2
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3
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304
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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
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0
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77
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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 ...
user111524
6
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1
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355
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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
2
votes
1
answer
323
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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
6
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1
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672
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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
2
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1
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232
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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
3
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1
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128
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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
7
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275
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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 ...
- 409
3
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1
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339
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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
4
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0
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280
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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
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1
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701
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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
2
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1
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138
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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
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1
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225
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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
1
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1
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141
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(...
user111524
1
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1
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787
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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 ...
user111524
1
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0
answers
138
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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 $?
- 21
13
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1
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940
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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
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0
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318
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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 ...
- 1,906
4
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1
answer
677
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$I,J$ 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
7
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0
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369
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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
2
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174
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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 ...
- 21
5
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1
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1k
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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
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1
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291
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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
1
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1
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258
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How to compute graph ideal or cut ideal of a graph?
Graph ideals are a special case of Stanley-Reisner ideal, explained in Combinatorial Commutative Algebra book by Sturmfels, and graph ideals here. Graph ideals are generated by the minimal paths while ...
- 143
1
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0
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91
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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 ...
- 143
5
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1
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1k
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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
2
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0
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134
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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. ...
- 2,232
2
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1
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2k
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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
21
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4
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5k
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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