All Questions
Tagged with ac.commutative-algebra ideals
31 questions with no upvoted or accepted answers
7
votes
0
answers
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
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
5
votes
0
answers
2k
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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
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
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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
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
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
2
votes
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
2
votes
0
answers
77
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 ...
user111524
2
votes
0
answers
174
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 ...
- 21
2
votes
0
answers
134
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. ...
- 2,232
1
vote
0
answers
205
views
Finding if an ideal is the radical of another one
Let's suppose we have, in the ring $\mathbb{Z} [x,y,z,w,v]$, the following polynomials:
$f=xw-yz$,
$g=x^2z-y^3$,
$h=yw^2-z^3$,
$k=xz^2-y^2w$.
The question is to prove that $I=(f,g,h,k)$ is the radical ...
1
vote
0
answers
85
views
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 $R$ ...
- 480
1
vote
0
answers
147
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 (...
- 807
1
vote
0
answers
108
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 ...
- 573
1
vote
0
answers
124
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....
- 2,089
1
vote
0
answers
140
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 ...
- 1,227
1
vote
0
answers
138
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 $?
- 21
1
vote
0
answers
318
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 ...
- 1,906
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 ...
- 143
0
votes
0
answers
92
views
Length of generic intersection in local ring
Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.
If $a$ that is not a zero divisor of $R/I$ we have ...
0
votes
0
answers
179
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 ...
- 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^...
- 1,615
0
votes
0
answers
105
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) \...
- 2,837