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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, \...
Simón Flavio Ibañez's user avatar
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2 votes
1 answer
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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 \...
D.R.'s user avatar
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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(...
Yoyo's user avatar
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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 ...
Sam Gue's user avatar
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5 votes
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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 ...
Math Lover's user avatar
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8 votes
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486 views

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 ...
Liam Baker's user avatar
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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 ...
Serge the Toaster's user avatar
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218 views

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 ...
Haze's user avatar
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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 ...
WittyCatchphrase's user avatar
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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 ...
A. Bailleul's user avatar
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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 $R$ ...
Alex's user avatar
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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 ...
user237522's user avatar
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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 $x^{-1}$...
Ralle's user avatar
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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 (...
Zhang Yuhan's user avatar
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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 $k\in\mathbb{N}$ ...
Sanae Kochiya's user avatar
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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 $...
Salvo Tringali's user avatar
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1 answer
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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 ...
Chess's user avatar
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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 ...
user237522's user avatar
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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}(...
user237522's user avatar
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1 vote
1 answer
217 views

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 $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle \subsetneq \mathbb{C}[x,y]$, ...
user237522's user avatar
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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 $...
user237522's user avatar
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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]/...
Sergiy Maksymenko's user avatar
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468 views

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[...
Jérémy Blanc's user avatar
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7 votes
1 answer
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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 \...
Samuel Johnston's user avatar
  • 534
3 votes
1 answer
305 views

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 ...
pinaki's user avatar
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220 views

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 ...
Ethan's user avatar
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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 ...
George Bentley's user avatar
  • 211
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1 answer
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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 ...
Mairon's user avatar
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144 views

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 ...
Varun Ramanathan's user avatar
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2 votes
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83 views

Are there any results related to properties of an ideal $I$ in $k[x_1,\ldots,x_n]$ generated by the polynomials $g(x_1,\ldots, x_m),\, g(x_2,\ldots, x_{m+1}), \ldots, g(x_{1+{n-m}},\ldots , x_{n})$? ...
olha's user avatar
  • 21
6 votes
1 answer
347 views

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, ...
darij grinberg's user avatar
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1 vote
1 answer
183 views

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 $k$-...
Boris's user avatar
  • 721
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1 answer
222 views

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}...
Andy's user avatar
  • 139
7 votes
2 answers
595 views

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\}...
Dominic van der Zypen's user avatar
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3 votes
1 answer
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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 $R/p$ ...
Cloudscape's user avatar
  • 597
2 votes
1 answer
207 views

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 $\pi$-...
Onur Oktay's user avatar
  • 3,008
3 votes
1 answer
308 views

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 $L(R)$ ...
Onur Oktay's user avatar
  • 3,008
2 votes
1 answer
202 views

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 ...
Oleksandr Kulkov's user avatar
  • 1,211
10 votes
2 answers
1k views

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 $\...
Elías Guisado Villalgordo's user avatar
  • 2,105
1 vote
2 answers
260 views

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) \...
Pintér Gergő's user avatar
  • 73
5 votes
0 answers
162 views

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 ...
Math Lover's user avatar
  • 1,105
5 votes
1 answer
256 views

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 ...
Wágner Badilla's user avatar
  • 217
0 votes
1 answer
204 views

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 \;\&\; ...
user65526's user avatar
  • 639
7 votes
1 answer
380 views

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 ...
Tomasz Kania's user avatar
  • 12k
3 votes
1 answer
520 views

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 ...
mrtaurho's user avatar
  • 165
2 votes
0 answers
154 views

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,...
Sylvain JULIEN's user avatar
  • 7,022
2 votes
1 answer
357 views

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 ...
darij grinberg's user avatar
  • 35.6k
4 votes
1 answer
298 views

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 ...
user avatar
2 votes
1 answer
302 views

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 ...
Math Lover's user avatar
  • 1,105
1 vote
1 answer
151 views

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)}=(...
Cusp's user avatar
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