All Questions
Tagged with ra.rings-and-algebras ac.commutative-algebra
667 questions
1
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1
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A question about surjective maps between quadratic algebras
Let $V$ be a finite-dimensional vector space and
$$
U \subseteq W \subseteq V \otimes V
$$
be a proper inclusion of vector subspaces. Then take the tensor algebra
$$
T(V) = \bigoplus_{i=1}^{\infty} V^{...
- 643
2
votes
0
answers
114
views
How many minimal relations are needed to obtain a Frobenius algebra?
Let $A_n:=K \langle x_1,x_2,...,x_n \rangle$ be the non-commutative polynomial ring in $n$-variables over the field $K$ and let $J=\langle x_1,...,x_n \rangle$ be the ideal spanned by the $x_i$.
An ...
- 26.5k
2
votes
2
answers
261
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Examples of stretched artinian local ring
In Sally's Paper stretched artinian local ring is defined as :
Let $(R, \mathfrak{m})$ be an Artin local ring of length $\lambda.$ If $\nu$ is the embedding dimension of $R$, that is, $\nu$ is the ...
- 81
3
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0
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225
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Intersection of two modules (and sub-modules) under tensors
I have a (hopefully quick) question regarding an intersection of two tensor modules. Let $K$ be a field and $A, B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...
- 31
4
votes
1
answer
295
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Finite type injective ring map between domains preserves the open point $(0)$
I am looking for a proof of the following statement without using full power of Chevalley's theorem on constructible sets. We say a domain $A$ is $0$-open if $\{(0)\}$ is open in $\operatorname{Spec}(...
- 143
4
votes
1
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370
views
Determining when quotient of a polynomial ring is a Gorenstein ring
I would like to be able to look at the ring $R=\mathbb{Z}[x_1,x_2,\ldots,x_n]/\mathcal{I},$ where $\mathcal{I}$ is generated by a finite number of monomials and say whether $R$ is a Gorenstein ring. ...
- 214
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0
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119
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Germs of holomorphic functions and invariant functions
Consider a complex vector space $V \cong {\mathbb C}^n$. Consider the ring of germs of holomorphic functions ${\mathcal O}_0 (V)$ at $0\in V$. We know that this ring is Noetherian.
Now consider a ...
- 803
4
votes
1
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141
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Kernels of actions on truncated polynomial algebra
Let $p$ be an odd prime, and let $k=\mathbb{F}_p$ be the field with $p$ elements. Let $G=\text{GL}_n(k)$. The group $G$ acts on the truncated polynomial algebra $A:=k[x_1,\ldots, x_n]/(x_1^p,\ldots, ...
- 5,039
-3
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1
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207
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can the square of a polynomial be written as a sum of squares of polynomials with only even degree terms?
Let $f$ be a polynomial with real coefficients in several indeterminates $x_1, \dots, x_n$. Suppose that
$$ f = g^2 $$
for some polynomial $g$.
Is it true that we can find polynomials $h_1, \dots, h_m$...
- 331
3
votes
1
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173
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$\Omega$ for noetherian semiperfect rings
Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-...
- 26.5k
3
votes
0
answers
119
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Finite algebras with finitely many automorphisms
Let $B'/B$ be a finite locally free algebra. Locally in $B$, there is an isomorphism of $B$-modules $B'\simeq B^{\oplus n}$. When is the automorphism group of $B'/B$ finite? When is it unramified? Is ...
- 1,084
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0
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172
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When does this commutative non-associative algebra have nilpotent elements?
Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients
$(a_0, \dotsc, ...
- 769
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114
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Inclusion between rings after localization
Let $\phi:A \to B $ an injective finite ring map between two noetherian integral domains $A,B$. Let $ C \subset B$ a subring of $B$ and assume that there exist a prime ideal $\mathfrak{p} \subset A$ , ...
- 5,998
1
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1
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155
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Invariant ring of the subvariety
Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also $...
- 839
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162
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Definitions of torch ring
Well, I find myself in a tangled web of references, definitions and doubt. How do I get myself into these things? Get ready for a rollercoaster of definitions.
An FGC ring is a commutative ring whose ...
- 1,507
0
votes
2
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387
views
Torsion of modules
Given a left module $M$ over a domain $R$, I am interested in irreducible elements $r\in R$ such that $r\cdot m=0$ for some $m\in M-\{0\}.$ I think "torsors" would be perfect name for such $...
- 2,390
3
votes
1
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382
views
What is the name for algebras generated by elements, all of whose cubes vanish?
Given a ring $R$ with identity $1$, we can define the exterior algebra of order $k$ over $R$ to be the algebra over $R$, generated by elements $x_1, \dots, x_k$ satisfying $x_i^2 = 0$ for each index $...
- 557
1
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2
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332
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Condition for equality of modules generated by columns of matrices
Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...
- 431
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149
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Rings with epimorphism from a finitely generated ring
For a commutative ring with unit $R$ let's say it has property $(*)$ if there is an epimorphism in the category of rings ${\mathbb Z}[X_1,\dots,X_n]\to R$, where the former is the polynomial ring in $...
user473423
0
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0
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267
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completion and tensor product
Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers?
If $A$ is noetherian, it is clear because one has for $k$ a residue ...
- 3,472
2
votes
1
answer
237
views
Is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{O}_{X})$-algebra?
Let $X$ be a scheme of finite type over $\mathrm{Spec}(A)$, where $A$ is a commutative ring. Let $U\subset X$ be any open subset, is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{...
- 23
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82
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Example of a secondary representation of a module that is not a direct sum
Let $A$ be a commutative ring. An $A$-module $M$ is said to be
secondary if $M\neq 0$ and for each $a\in A $, the endomorphism
$\phi_a:M\to M$ defined by $\phi_a(m)=am$ for $m\in M$ is either
...
- 631
1
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0
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66
views
Existence of a minimal ideal with a specific property
Suppose that $R$ is a super-commutative ring (i.e. it is a unital $\mathbb{Z}_2$-graded ring satisfying $xy=(-1)^{|x|\cdot |y|}yx$ where $|x|$ denotes the grading degree of a homogeneous element $x\in ...
- 329
3
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1
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420
views
What is a PBW algebra? (I.e., an algebra generalising properties of $U(\frak{g})$)
I am reading a paper where they refer to a certain algebra as a PBW algebra. What does this mean exactly? I would infer from the $U(\frak{g})$ setting that this means the existence of an ordered ...
2
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0
answers
198
views
Ring homomorphisms from the commutative ring into $\mathbb{Z}_2$
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Spec{Spec}$Let $A$ be a commutative ring not necessarily with unit and $\mathbb{Z}_2 =\{0,1\}$ be the field of two elements. I am looking for a paper ...
1
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0
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60
views
Size of minimal generating set of a module generated by columns of a diagonal matrix with extra structure
Let $R$ be a commutative ring with unit. Let $A \in R^{k \times k}$ be a diagonal matrix such that $A_{11} | A_{22} | \dots | A_{rr}$ for some $r \leq k$ and are non-zero, while $A_{ii}=0$ for all $i &...
- 431
5
votes
2
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444
views
For a finite-type $\mathbb{Z}$-algebra $A$, is the intersection of all ideals $I$ such that $A/I$ is finite and local necessarily zero?
Background: A referee has suggested a shorter proof of one of my results, but I'm having trouble justifying one of their assertions. The setting is that $A$ is a commutative ring, and the referee's ...
- 2,356
4
votes
1
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164
views
Any ideal as an intersection of ideals primary to maximal ones
The Nullstellensatz says that any ideal $I \subset \mathbb{C}[x_1,\dots,x_n]$ has the property that
$\sqrt{I} = \bigcap_{\text{maximal } \mathfrak{m} \supset I} \mathfrak{m}$
Is it also true that we ...
- 41
28
votes
2
answers
863
views
$A^2$ is isomorphic to $A^{(\omega)}$, but not $A$
Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)?
Edited to add: As no answers are forthcoming, does anyone ...
- 18.7k
2
votes
1
answer
194
views
Structure of reflexive modules over regular local rings
Let $R$ be a regular local ring and $M$ a finitely generated reflexive $R$-module. When $R$ has dimension 2, then $M$ is a free $R$-module. This is discussed in Reflexive modules over a 2-dimensional ...
- 429
3
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0
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67
views
Atomicity and BF-ness in monoids of integer points of a polyhedral cone of $\mathbb R^n$
Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is ...
- 10.5k
4
votes
0
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216
views
Characterizing atomicity in a commutative domain
In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ...
- 10.5k
1
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0
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329
views
Outlier absences of monomials in a group of inversion partition polynomials
Revamped and updated on Sep 12, 2022:
Given the complex coefficients $a_n$ of some generic formal power, Taylor, Laurent or other series, say the ordinary generating functions (o.g.f.) $f(z) = z +a_1 ...
- 10.5k
7
votes
1
answer
408
views
Homological dimensions of rings of smooth functions
What is the global dimension of the algebra $C^\infty\mathbb R$ of smooth functions $\mathbb R\to\mathbb R$? What is the global dimension of the algebra $(C^\infty\mathbb R)_0$ of germs of smooth ...
- 700
5
votes
2
answers
236
views
An example of a local integral domain with special spectrum
I am looking for a local integral domain $(D, m)$ with $Spec(D)=\{0,m\}\cup\{ P_i\}_i$ such that $P_i's$ are incomparable (that is, $P_i\not\subseteq P_j$ and $P_j\not\subseteq P_i$ for $i\not= j$) ...
- 147
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2
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789
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Algebraic power series of finite order
Apologies if the question is too elementary/something well-known.
I believe it is a well-known fact that the rational formal power series $F(z)=\frac{P(z)}{Q(z)}$ which have finite order under ...
- 24.2k
7
votes
1
answer
271
views
Algebraic proof that the monoid ring of a torsion-free monoid is reduced
In what follows, I say that a monoid $M$ is torsion-free if the $n$-th power map is injective for all $n \geq 1$. I have a proof of the following result:
Claim: if $M$ is a torsion-free commutative ...
- 413
1
vote
1
answer
152
views
Cohen-Macaulay quotient ring and symbolic power
Let $(R, \mathfrak{m})$ be a regular local ring and let $\mathfrak{a} \subset R$ be an ideal. Let
$$ \mathfrak{b} = \bigcap \{R \cap \mathfrak{a} \cdot R_\mathfrak{p} \text{ } \colon \mathfrak{p} \in \...
3
votes
4
answers
807
views
$R$ is a UFD iff $R_{\frak{m}}$ is a UFD?
Let $R$ be a graded ring such that $R_0$ is a field and let $\frak{m}$ be the maximal ideal generated by all the elements of positive degree. Then, is it true that $R$ is a UFD iff $R_{\frak{m}}$ is a ...
- 839
1
vote
0
answers
87
views
When is the product of two elements in algebraic closures of rational functions a constant function?
I have one question on some interactions between sum and product of elements in algebraic clsoures of rational polynomials over algebraically closed fields.
My question is as follows:
Let E and F be ...
- 11
5
votes
1
answer
317
views
Localization of a ring and the Hom functor
Let $R=\mathbb{Z}[x,x^{-1}]$ be the ring of Laurent polynomials in $x$, $\mathfrak{p}=(1-x)$ be an ideal in $R$ and $R_\mathfrak{p}$ be the localization. I want to know what $\text{Hom}_R(R_\mathfrak{...
user340793
2
votes
0
answers
82
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Let $R$ be a non-catenary, and $f: R \to S$ be a finite monomorphism. Can $S$ be catenary?
Let $R$ be a commutative ring with identity. Then $R$ is $\textit{catenary}$ if for each pair of prime ideal $p \subsetneq q$, all maximal chains of prime ideals $p = p_0 \subsetneq p_1 \subsetneq \...
- 1,355
0
votes
0
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93
views
In $\mathbb{Z}[G]$, $G\cong \mathbb{Z}^r$, does $f\cdot g\geq 0$ imply $f\geq0$?
Let $G=\mathbb{Z}^r$ be a free abelian group, and $\mathbb{Z}[G]$ be the group ring of $G$. Define a partial ordering $\leq$ on $\mathbb{Z}[G]$ by
$$\sum_{g\in G}n_g[g]\leq\sum_{g\in G}n'_g[g]\iff n_g\...
- 2,902
0
votes
0
answers
228
views
Generalization of "Lagrange interpolation" over non-division rings
The theorem below is from pages 4 and 5 in Singmaster - On polynomial functions $\pmod m$ (Theorem 10) on polynomials in $\mathbb{Z}_m[x]$.
Let $f$ be a polynomial function $\pmod{m}$. Then $f$ has a ...
- 11
1
vote
0
answers
123
views
Clarification about theorem on vanishing polynomials
The theorem below is from page 3 in the this paper on polynomials in $\mathbb{Z}_m[x]$.
Let $F$ be a polynomial in $\mathbb{Z}_m[X]$. Then $f \equiv 0$ iff $$F \equiv F_nS_n + \sum_{k=0}^{n-1}a_k(m/(...
- 11
5
votes
2
answers
341
views
Writing $1-xyzw$ as a sum of squares
Can you write $1 - xyzw$ in the form $p + q (1 - x^{2}-y^{2}-z^{2}-w^{2})$ where $p$ and $q$ are polynomials that are of the form $\sum g_{i}^{2}$ where $g_{i}$ $\in$ $\mathbb{R}[x,y,z,w]$?
For ...
5
votes
1
answer
335
views
About the structure of unit groups appearing in number theory
I think the following statement is not true in the general situations, but consider it:
$R$ is a ring, $\mathfrak{p}$ is a prime ideal, then the unit group of $\dfrac{R}{\mathfrak{p}^nR}$ is ...
5
votes
1
answer
420
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Ring of continuous functions is a Jacobson ring
Let $X$ be an infinite discrete topological space. Is $$C_b(X)=\{ f \colon X \to \mathbb{R} \text{ bounded }\}$$ a Jacobson ring ?
9
votes
0
answers
460
views
Does the book "Algebra III" exist (within the Encyclopaedia of Mathematical Sciences series from Springer)?
Within the series "Encyclopaedia of Mathematical Sciences", as published by Springer, one finds the 8 volumes, namely,
the volumes I, II, IV, V, VI, VII, VIII, IX but zbMath has no listing ...
- 161
2
votes
0
answers
80
views
Prime elements in integrally closed extension of domains
Let $R \subseteq S$ be an extension of integral domains such that $R$ is integrally closed in $S$. Let $P$ be a prime ideal of $R$.
Is $PS$ always a prime ideal of $S$?
For classical examples of ...
- 365