All Questions
Tagged with rings-and-algebras or ra.rings-and-algebras
3,500 questions
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Is the multiplication between even numbers an associative algebra? [closed]
We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist?
It has been proposed as a counterexample the set of even numbers. ...
6
votes
1
answer
455
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Mechanically instantiating abstract constructions
I am looking for work on the effective inverse of abstraction, aka specialization.
There are two ways in which abstraction helps us:
Get a better understanding of the structural rules at play in ...
- 11.8k
4
votes
0
answers
325
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Localization of power series and module structure
Let $R=\mathbb{Q}[X,Y]$ be the polynomial ring of two commuting variable.
Let $S$ be the multiplicative subset of $R$ generated by homogeneous linear polynomials.
Let also $\widehat{R}$ be the ring of ...
- 41
23
votes
6
answers
2k
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Are rings really more fundamental objects than semi-rings?
The discovery (or invention) of negatives, which happened several centuries ago by the Chinese, Indians and Arabs, has of course be of fundamental importance to mathematics.
From then on, it seems ...
9
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3
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629
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Terminology: Algebras where long strings of products are 0?
I'd like a name for an augmented algebra $A = \langle 1\rangle \oplus A_+$ for which there is an $N$ so that any product of more than $N$ elements in the augmentation ideal is $0$, i.e., $(A_+)^N = 0$....
- 10.1k
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2
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619
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Problems concerning R and R[x]
A few questions relevant formally, but quite different in nature:
From now on, let R denote a ring.
If R is a UFD , is R[x] also a UFD?
If R is Noetherian, is R[x] also Noetherian?
If R is a PID, ...
- 363
5
votes
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388
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is there a notion of weakly noetherian?
A left module M over a ring is finitely presented iff the functor Hom(M,_) preserves filtered inductive limit. Meanwhile, if M is finitely generated, then for every filtered system { N_i } of left ...
- 403
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3
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3k
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Dimension of subalgebras of a matrix algebra
If $n$ is given and $A$ is a subalgebra of $M_n(\mathbb C)$, the algebra of $n \times n$ matrices with entries in the field of complex numbers, then what are the possible values of dimension of $A$ as ...
- 181
1
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1
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310
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How to distinguish property of particular representation from property of algebraic structure?
It is common that you have some interesting object (set, group, algebra or something, whatever) which has certain properties, structure etc. You may try describe it in pure algebraic way. Sometimes ...
- 1,626
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1
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820
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Does a finite dimensional algebra having a Cartan matrix with determinant 1 imply finite global dimension (possibly with more hypotheses)?
Let $A$ be your favorite finite dimensional algebra, and $P_i$ be a sets of representatives for the indecomposible projectives (or PIMs, if you like). Then we have the Cartan matrix $C$ of the ...
- 44.7k
2
votes
1
answer
245
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Characterization of a certain class of modules-broader than Noetherian
Let $R$ be a commutative ring with $1$.
An $R$-module $K$ has the 'S' property if $K/T \simeq K$ (i.e. isomorphic) implies that the submodule $T$ is trivial.
By Fitting's lemma any Noetherian module ...
- 143
1
vote
1
answer
736
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F is ultrafilter over a Boolean algebra implies that for every b, either b or not-b is in F?
I'm trying to teach myself category theory from Steve Awodey's Category Theory. Chapter 2 asserts:
It is not hard to see that a filter F is an ultrafilter just if for every element b ∈ B, either b ∈...
- 150
18
votes
3
answers
1k
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Are there countable index subrings of the reals?
Does ${\mathbb R}$ have proper, countable index subrings? By countable I mean finite or countably infinite. By subring I mean any additive subgroup which is closed under multiplication (I don't care ...
- 691
15
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1
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2k
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Is a left invertible element of a group ring also right invertible?
Given a group $G$ we may consider its group ring $\mathbb C[G]$ consisting of all finitely supported functions $f\colon G\to\mathbb C$ with pointwise addition and convolution. Take $f,g\in\mathbb C[G]$...
- 637
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4
answers
4k
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Example of the completion of a noetherian domain at a prime that is not a domain
Let $R$ be a Noetherian domain, and let $\mathfrak{p}$ be a prime ideal; consider the completion $\hat R_{\mathfrak{p}}$ of $R$ at $\mathfrak{p}$ (the inverse limit of the system of quotients $R/\...
- 7,207
3
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271
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Algebraic Kneser conjecture?
Recall that Kneser conjecture (now Lovasz theorem) claims that if the family of $k$-subsets (subsets of cardinality $k$) of given $(2k+d)$-set $M$, $d\geq 1$ are colored into $d+1$ colors, then there ...
5
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0
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518
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Generalized Haar Measures and Semiring-Valued Integrals on Lie Groups
In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete ...
- 2,392
21
votes
3
answers
2k
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Is there a "categorical" description of Grothendieck's algebra of differential operators?
First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due ...
- 54.6k
6
votes
7
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5k
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Best way to teach concept of real numbers using a hands-on activity?
I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.
- 163
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1
answer
494
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Tensor products and two-sided faithful flatness
Let $f: R \to S$ be a morphism of Noetherian rings (or more generally $S$ can just be an $R-R$ bimodule with a bimodule morphism $R \to S$). Suppose $f$ is faithfully flat on both sides, so $M \to M \...
- 37k
2
votes
1
answer
757
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Maximal subfield inside a central division algebra
D is a central division algebra over F. We know that we can always find a maximal subfield K inside D such that K/F is separable. I want to know can we always make it Galois?
- 1,091
3
votes
1
answer
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Equational definition of Residuated Lattices
The usual axiomatization of residuated lattices involve using ≤. I know I can expand away the use of ≤ using a definition such as (x ≤ y) := (x ∧ y = x), but I fear I will get a set of messy axioms.
...
- 1,439
62
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25
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70k
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Linear Algebra Texts?
Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...
3
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1
answer
336
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Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$.
The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$
has the bracket
$$[xt^r, yt^...
- 138
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1
answer
633
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Introduction to "commutative semialgebra"?
Of course, commutative algebra is a fundamental topic in algebraic geometry, number theory, representation theory, and so on.
However, there are some instances (most obviously tropical geometry) ...
- 12.6k
7
votes
1
answer
8k
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When are the units of R[x] exactly the units of R?
I (Anton) have edited this question to be the question Pete and Zeb discuss in the first few comments.
What conditions on a ring $R$ imply that the units of $R[x]$ are exactly the units of $R$?
- 119
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2
answers
6k
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Using Weierstrass’s Factorization Theorem
I am trying to factorize $\sin(x)\over x$ which by Taylor series expansion and using the roots is $$a \cdot \left(1 - \frac{x}{\pi} \right) \left(1 + \frac{x}{\pi} \right) \left(1 - \frac{x}{2\pi} \...
- 5,935
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1
answer
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LUB and GLB on a lexicographically ordered complete lattice product
I am trying to define the LUB and GLB on a product of lattices that are partially ordered lexicographically.
Is there any papers or help that I could read up on? I would particularly like proofs on ...
- 71
3
votes
3
answers
213
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What can I say for free about a module with dominant dimension 2 (other than the double centralizer property)?
Let's say I have my favorite finite dimensional algebra $A$, and favorite module $T$. Now assume that the reason $T$ is my favorite module is that it has a cool property:
there is an injection $A\...
- 44.7k
5
votes
2
answers
752
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Is there a name for this algebraic structure?
I found myself "naturally" dealing with an object of this form:
X is a complex vector space, with a "product" (a,b) → {aba} which is quadratic in the first variable, linear in the second, and ...
- 971
6
votes
3
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2k
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Constructing a metric over a lattice
Consider a lattice $({\cal L}, \wedge, \vee)$ with an antimonotonic function $f: {\cal L} \rightarrow {\mathbb R}$ defined on it (i.e $x \preceq y \implies f(x) \ge f(y)$).
$f$ is said to be ...
- 4,462
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votes
3
answers
2k
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Is there a version of the Archimedean property which does not presuppose the Naturals?
All the statements of the Archimedean property with which I am acquainted fundamentally uses ℕ -- more than as a totally ordered semi-group, really being the 'standard model' of the naturals. ...
- 11.8k
4
votes
2
answers
920
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What is the correct formulation of the CDE triangle?
The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as ...
- 9,132
9
votes
2
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984
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Spheres over rational numbers and other fields
Let K be an ordered field. Define the n-sphere:
$$S^n(K) := \{ (x_1,x_2,\dots,x_n+1) \in K^{n+1} \mid \sum_{i=1}^{n+1} x_i^2 = 1 \}$$
A set of vectors $v_1, v_2, \dots, v_r \in S^n(K)$ is ...
- 7,320
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votes
2
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2k
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Examples of one-dimensional non-Cohen Macaulay rings
Can you offer some examples of such rings, other than $\frac{k[x,y]}{(x^{2}, xy)}$. Thanks.
- 113
2
votes
2
answers
718
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Algebra / unital associative algebra: better terminology?
In Bourbaki an algebra over a commutative ring $k$ is defined to be a $k$-module $A$ together with a $k$-bilinear map $A \times A \rightarrow A$. We then have the obvious notion of morphisms of $k$-...
1
vote
1
answer
146
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equivalence of submodules
I have Z^3/M = Z^3/N = Z_k where M,N are submodules of Z^3 and Z_k is cyclic order k.
I would like to say some SL_3(Z) transformation takes M to N. Is this true? How to show?
- 560
2
votes
2
answers
423
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characterization of a submodule
In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if it is closed under addition and scalar multiplication. For a module $M$ over a ring $R$ with identity the similar ...
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2
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259
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Existence of an "anti-additive" (or "never linear") map?
(I've edited this question)
I'm searching for a continuously differentiable function $f:\mathbb R^2\to\mathbb R$ such that $f(x)+f(x+u+v)\neq f(x+u)+f(x+v)$ for all $x$ and all linearly independent $...
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1
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335
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How many rings exist when ring is subspace of finite dimensional vector space?
Suppose we have a ring $R[M]$ for a monoid $M$ over the real numbers $R$. The number of generators for the monoid is finite. Now suppose that every ring element $r$ has a decomposition in finite ...
- 1,626
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1
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First-order UFD (factorial ring) condition / pre-Schreier rings
All rings in this post are commutative and with $1$.
Everyone knows the definition of a factorial ring, a. k. a. unique factorization domain (UFD). I have been wondering about some variations ...
- 33.8k
53
votes
7
answers
14k
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Good lattice theory books?
A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - ...
52
votes
7
answers
8k
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"Algebraic" topologies like the Zariski topology?
The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...
2
votes
1
answer
1k
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monoid ring and some structure within it - how is it called?
I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure ...
- 1,626
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vote
2
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Relations in matrix semigroups
Suppose that $A_1, \dots, A_k \in M_n(\mathbb{Q})$ and $S$ is the semigroup generated by them. Two questions: are there always a finite set of relations $\{R_i\}$ among the $A_j$ such that $S$ is ...
- 4,636
4
votes
3
answers
388
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Characterizing nilpotents in a ring by a universal property
This is again a question asked to me by this user. He apparently quit using MO due to a busy time in personal and professional life and resulting difficulties in spending time here with patience. I am ...
- 7,442
1
vote
2
answers
865
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Cayley-Dickson form of a quaternion
It is known that using the Cayley-Dickson construction a quaternion $q$ can be written in a symplectic form as $q=x+\mathbf{i}y$ with $x,y \in \mathbb{C}$.
I read in a couple of references that $x$ is ...
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2
answers
1k
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Is the matrix ring $\mathrm{Mat}_n(\mathbb{C})$ "algebraically closed"?
In spite of the fact that the matrix ring $\mathbb{C}^{n \times n}$ is not a field, is it still possible to talk about it being 'algebraically closed' in the sense that $\forall f \in \mathbb{C}^{n \...
- 3,165
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votes
5
answers
1k
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Which R-algebras are the group ring of some group over a ring R?
This question was inspired by Qiaochu's recent question, Which commutative groups are the group of units of some field? - my question is close to being the inverse of it.
As mentioned here, given a ...
- 6,792
7
votes
3
answers
1k
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Intersection of finitely generated subalgebras also finitely generated?
Let $k$ be a field and $A$ be a finitely generated (commutative) algebra over $k$. If $A_1$ and $A_2$ are finitely generated $k$-subalgebras of $A$, is it true that $A_1 \cap A_2$ is also finitely ...
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