Questions tagged [ideals]
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153 questions
4 votes
1 answer
499 views
Maximal ideals in rings with polynomial relations
Let $F$ be a perfect field, and let $p(t) \in F[t]$ be an irreducible monic polynomial of degree $n$ such that: $$p(t) = t^n+s_1 t^{n-1}+s_2 t^{n-2}+\dots+s_n$$ Let $\theta_1, \theta_2, \dots, \...
2 votes
1 answer
170 views
Geometric/linear algebra proof of multiplicativity of ideal norm in ring of integers
For a number field $K/\mathbb Q$ (as a $\mathbb Q$-vector-space $V$, $n$-dimensional), we have the ring of integers $\mathscr O_K$ (a lattice = copy of $\mathbb Z^n$ in $V$). Ideals $I \subseteq \...
- 1,245
3 votes
1 answer
276 views
Classes of "bases" of an ideal of polynomials
Let $k$ be a field and $I$ an ideal of $k[x,y]$ generated by two polynomials. Let $\mathcal{E}$ be the set defined by $$ \mathcal{E}=\{(P,Q)\in k[x,y]^2 | \langle P,Q\rangle =I\}.$$ The group $G=GL_2(...
- 321
1 vote
1 answer
119 views
If a parametric polynomial system is zero-dimensional, then is the specialized polynomial system zero-dimensional for almost all parameter values?
If we have a parametric polynomial system in say $Q = \mathbb{Q}[p_1,\ldots,p_m][x_1,\ldots,x_n]$ and $I$ is zero-dimensional over $\overline{\mathbb{Q}[p_1,\ldots,p_m]}$, then is it true that for ...
- 21
5 votes
2 answers
232 views
Comparing two topologies on the space of ideals of a $C^{\ast}$-algebra
Let $A$ be a $C^*$-algebra and consider two topologies on $\text{Id}(A)$, the set of all ideals of $A$. In the first topology, a basis is given by sets of the form $$U(K) = \{ I \in \text{Id}(A) \mid ...
- 1,105
8 votes
1 answer
472 views
Rings where each left principal ideal is also a right principal ideal
I'm interested in rings $R$ where for each $a,b$ in $R$ there is a $c \in R$ such that $ab = ca$, or equivalently where $aR = Ra$ for each $a \in R$. Obviously commutative rings and skew fields ...
- 425
0 votes
0 answers
126 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 ...
4 votes
0 answers
216 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
1 vote
0 answers
271 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
1 answer
209 views
Quotient rings of integral quaternion rings
I'm having a hard time finding information about the quotient rings of the Lipschitz quaternions and the Hurwitz quaternions. The Lipschitz quaternions are defined as the quaternions with integral ...
- 1,352
1 vote
0 answers
107 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
4 votes
0 answers
283 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 ...
- 2,875
1 vote
0 answers
75 views
Ideals of Laurent polynomial ring over matrix ring
Let $K$ be a field. Let $R=M_2(K)\langle x,x^{-1}\rangle$ be the ring obtained from the matrix ring $M_2(K)$ by adjoining two elements $x$ and $x^{-1}$ which are inverse to each other ($x$ and $x^{-1}$...
- 491
1 vote
0 answers
183 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 (...
- 1,007
1 vote
0 answers
335 views
Is the span of all nilpotent ideals also a nilpotent ideal?
Given a non-zero Lie algebra $\mathcal{L}$ over $\mathbb{C}$, we define $\mathcal{L}^2 = \big[\mathcal{L}, \mathcal{L} \big] = \big\{ [x, y]: x, y\in \mathcal{L} \big\}$, and for any $k\in\mathbb{N}$ ...
- 369
3 votes
0 answers
129 views
An isomorphism problem for semigroups of ideals
An ideal of a semigroup $S$ (written multiplicatively) is a set $I \subseteq S$ such that $IS$ and $SI$ are both contained in $I$ (here, $XY$ means, for all $X, Y \subseteq S$, the setwise product of $...
- 11.4k
1 vote
1 answer
209 views
Gorenstein property from initial ideal
My question is: If $I$ is a homogenous ideal of $S=K[x_1,\dots,x_n]$ and $\mathrm{in}_{<}(I)$ is the initial ideal of $I$, with respect to a term order $<$ on $S$, then $S/I$ is Gorenstein if ...
- 1,357
1 vote
1 answer
176 views
If $(f,g)$ and $(f,h)$ are maximal ideals, then $ag+bh=P(f)$ for some $a,b \in k, P(t) \in k[t]$?
Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$. Let $f,g,h \in k[x,y]$, $g \neq h$, satisfy the following two conditions: (1) $(f,g)$ is a maximal ideal of ...
- 2,875
0 votes
1 answer
168 views
$k(F_i)_{i=1}^{n}=k(G_j)_{j=1}^{m}$ iff there exist $a_i,b_j \in k$ such that $\langle F_i-a_i \rangle_{i=1}^{n} = \langle G_j-b_j \rangle_{j=1}^{m}$
Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$ and let $F_1,\ldots,F_n,G_1,\ldots,G_m \in \mathbb{C}[x,y]$, $n,m \in \mathbb{N}-\{0\}$. Claim: $\mathbb{C}(...
- 2,875
1 vote
1 answer
211 views
Ideals: If $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle$, then $\langle f_1-\lambda,f_2-\mu \rangle = \langle g_1-\delta,g_2-\epsilon \rangle$?
The following question appears in MSE without answers. Let $f_1,f_2,g_1,g_2 \in \mathbb{C}[x,y]-\mathbb{C}$. Assume that $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle \subsetneq \mathbb{C}[x,y]$, ...
- 2,875
1 vote
1 answer
207 views
$\mathbb{C}(u(x,y),v(x,y),f(x)+g(y))=\mathbb{C}(x,y)$ implies $\mathbb{C}(u(x,y),v(x,y))=\mathbb{C}(x,y)$?
The following question is a direct continuation of this question: Let $u,v \in \mathbb{C}[x,y]$. Assume that for every $f \in \mathbb{C}[x]$ and every $g \in \mathbb{C}[y]$ (excluding the cases where $...
- 2,875
0 votes
0 answers
173 views
Is that true that each ideal $I \subset k[x_1,\ldots,x_n]$ of finite $k$-codimension contains all monomials of sufficiently high degree?
Let $k$ be a field, and $I$ be an ideal in the $k$-algebra $k[x_1,\ldots,x_n]$ of all polynomials of $n$ variables. Suppose that $I$ has finite codimension over $k$, i.e. $$ \dim_{k} k[x_1,\ldots,x_n]/...
2 votes
1 answer
463 views
What is the ideal of hypersurfaces singular at a given irreducible variety?
Let $X\subseteq \mathbb{P}^n$ be a closed irreducible subvariety, with vanishing ideal $I(X)\subseteq k[x_0,\ldots,x_n]$, where $k$ is the ground field, assumed to be algebraically closed. Let $F\in k[...
- 7,916
7 votes
1 answer
378 views
The combinatorics of the Nullstellensatz for the variety of nilpotent matrices
Let $H_n$ denote the set of $n \times n$ nilpotent matrices with complex entries. The set $H_n$ may be regarded as an algebraic variety. Indeed, consider the polynomial ring $\mathbb{C}[A_{i,j} : 1 \...
- 534
3 votes
1 answer
302 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 ...
- 5,479
0 votes
0 answers
219 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
1 vote
0 answers
75 views
Natural Density of Norms of ideals in a given ideal class
Some time ago, Landau proved the following formula for general number fields: $I_K(x)=U_Kx+O(x^{\delta})$, where $\delta=1-2/(1+[K:\mathbb{Q}])$, where $I_K(x)$ is the number of ideals with norm below ...
- 211
0 votes
1 answer
299 views
Are zero dimensional ideals radical?
I have a question about Theorem 3.7.25. of Computational commutative algebra I by M. Kreuzer and L. Robbiano. Let $K$ be a perfect field, $I \subseteq K[x_1, \ldots, x_n]$, be a zero dimensional ...
- 131
4 votes
0 answers
142 views
Which projections maintain irreducibility of the polynomial $x_0x_1 + x_2x_3 + \dots + x_{2m-2}x_{2m-1}$?
Let $q = x_0x_1 + x_2x_3 + \dots + x_{2m-2}x_{2m-1} \in \Bbb{C}[x_0, \dots, x_{2m-1}]$. The ambient space is $\Bbb{C}^{2m}$. Which are the polynomials $p \in \Bbb{C}[x_0, \dots, x_{2m-1}]$ such that ...
2 votes
0 answers
83 views
Ideals by the polynomial with "shifted" variables like g(x,y,z,) g(y,z,u), g(z, u,v)
Are there any results related to properties of an ideal $I$ in $k[x_1,\ldots,x_n]$ generated by the polynomials $g(x_1,\ldots, x_m),\, g(x_2,\ldots, x_{m+1}), \ldots, g(x_{1+{n-m}},\ldots , x_{n})$? ...
- 21
6 votes
1 answer
343 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, ...
- 35.5k
1 vote
1 answer
181 views
How to compute the associated reduced ring for this finitely generated algebra?
Let $k$ be a field, $m$ be a positive integer and $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x,y]$. Let $B$ be the quotient ring $R/xR$. Then $B$ is the finitely generated $k$-...
- 721
0 votes
1 answer
222 views
Are the ideals in two $C^*$-algebras the same?
Let $V_{1}, V_{2}$ be the commuting isometries. By Wold decomposition theorem, we know that $V_{i}$ admits decomposition $$V_i \cong V^s_{i}\oplus V^{u}_{i},$$ where $V^{s}_{i}$ is the shift and $V^{u}...
- 139
7 votes
2 answers
583 views
Counterexample for Chvatal's conjecture in an infinite set
Let $X \neq \emptyset$ be a set. We say that ${\cal F} \subseteq {\cal P}(X)$ is a down-set if ${\cal F}$ is closed under taking subsets. Whenever $a \in X$, we let ${\cal F}_a = \{ S \in F : a \in S\}...
- 53.7k
3 votes
1 answer
288 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 $R/p$ ...
- 597
2 votes
1 answer
206 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 $\pi$-...
- 2,978
3 votes
1 answer
306 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 $L(R)$ ...
- 2,978
2 votes
1 answer
200 views
Hadamard product of linear recurrences with umbral calculus
Let $R$ be a ring, $d_0, d_1, d_2, \dots \in R$ and $e_0, e_1, e_2, \dots \in R$ be linear recurrence sequences, such that $d_m = a_1 d_{m-1} + a_2 d_{m-2} + \dots + a_k d_{m-k}$ for $m \geq k$, $e_m ...
- 1,211
10 votes
2 answers
1k views
Is the ideal product presheaf a sheaf? Do we have any reasons to believe it will be / it won't?
My question is about one of those several concepts in algebraic geometry who everybody uses but nobody defines or introduces properly. Given a ringed space $(X,\mathcal{O}_X)$ and ideal sheaves $\...
1 vote
2 answers
254 views
The quotient of an algebra with an ideal whose generators are decomposed as the product of irreducible elements
I would like to find reference for the following statement. I need it only in the particular case when $A=\mathcal{O}_{(\mathbb{C}^n, 0)}$ is the local algebra of holomorphic germs $(\mathbb{C}^n, 0) \...
- 73
5 votes
0 answers
159 views
Trying to prove a seemingly easy fact on ideals of ternary C*-algebras
Currently I'm reading the paper by Abadie and Ferraro titled Applications of ternary rings to $C^*$-algebras. Recall that a $C^{\ast}$-ternary ring is a complex Banach space $M$, equipped with a ...
- 1,105
5 votes
1 answer
255 views
Monomial order and prime ideals
Let $S=K[x_1,\ldots, x_n]$ polynomial ring. Let $I \subseteq S$ an ideal and $<$ be a global monomial order in $S$. Suppose that $\mathrm{in}_<(I)$ is a radical ideal. Is it possible to describe ...
- 217
0 votes
1 answer
196 views
Partial orders on downward closed sets [closed]
Let $P = (V, \sqsubseteq)$ be a partial order and $\mathfrak{D}(P)$ denote the class of downward-closed subsets of the partial order $P$ (i.e, the class of $A \subseteq V$ such that $y\in A \;\&\; ...
- 639
7 votes
1 answer
376 views
Does every infinite-dimensional Banach algebra contain an infinite-dimensional subalgebra with second-countable primitive ideal space?
Let $A$ be an infinite dimensional Banach algebra. Even if separable the primitive ideal space of $A$ need not be second-countable when endowed with the hull-kernel topology. Can we at least find an ...
- 11.7k
3 votes
1 answer
509 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
2 votes
0 answers
154 views
Quasi-ideals and Erdős conjecture on arithmetic progressions
Starting to read a book about algebraic geometry as well as the Wikipedia article on Erdős conjecture on arithmetic progressions, I came to think of what follows. Let $A$ be a set of positive integers,...
- 7,016
2 votes
1 answer
355 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 ...
- 35.5k
4 votes
1 answer
297 views
$A$ is a commutative connective dg-algebra satisfying $H^0(A)=k$. Is it true that a dg ideal generated by elements of negative degree acyclic?
Let $k$ be a field of characteristic zero and $A$ be a graded commutative dg-algebra over $k$ with differential of degree $+1$ satisfying $H^0(A)=k, H^i(A)=0$ for $i<0$. Denote by $\mathcal J$ a dg ...
user21167
2 votes
1 answer
299 views
Primitive ideals of minimal tensor product
Let $A$ and $B$ be $C^{\ast}-$ algebras and $A \otimes B$ denotes minimal(spatial) tensor product. Is there any classification of primitive ideals of $A \otimes B$? (I'm mainly interested in the ...
- 1,105
1 vote
1 answer
149 views
Symbolic power of an ideal associated to non-singular algebraic set
Let $Z\subset \mathbb P^n$ be a reduced non-singular algebraic set and $I$ denote the saturated homogeneous ideal of $Z$. I have seen the following result without proof: For all $ n\geq 1$, $I^{(n)}=(...
- 1,783