Questions tagged [ra.rings-and-algebras]
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
3,364
questions
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Which definitions of "local module" have gotten traction?
It seems like "local module" has been defined a lot of ways:
if 𝑀 has a largest proper submodule. (This math.se post)
if it is hollow and has a unique maximal submodule (Singh, Surjeet, ...
- 1,603
2
votes
0
answers
97
views
Is the associated algebra of a quadratic filtered algebra again quadratic
Let $A$ be a filtered quadratic algebra. Let $G(A)$ be the associated graded algebra. Will $G(A)$ again be a quadratic algebra or can higher relations appear when passing to the graded seeting?
EDIT: ...
1
vote
1
answer
123
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Two (or less) filtrations on coenveloping coalgebra
Conilpotent coenveloping coalgebra UC(T) of a conilpotent Lie coalgebra T is defined by an universal property, similar to usual enveloping algebra: it's a coassocative, conilpotent coalgebra UC(T) ...
- 4,436
2
votes
1
answer
129
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Coproduct for a Frobenius algebra
The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ...
3
votes
1
answer
359
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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 ...
3
votes
0
answers
172
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Where could a paper on a unification of matrix decompositions be published?
I've got a paper which shows that when the spectral theorem (as a statement that every self-adjoint matrix can be unitarily diagonalised) is naively generalised to $*$-algebras other than the complex ...
- 4,843
4
votes
0
answers
66
views
Real forms of complex Lie algebras with specified semisimple part
Let $\mathfrak g$ be a complex Lie algebra. If $\mathfrak g$ is semisimple then its real forms are completely understood. In particular there is a compact real form of $\mathfrak g$, which is unique ...
- 4,674
2
votes
0
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183
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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 ...
7
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1
answer
190
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Free median algebras and maximal linked systems
$\DeclareMathOperator\MLS{MLS}$Recall that the median operation, on the power set $2^Y$ of subsets of a set $Y$, is the ternary law $m(A,B,C)$ mapping a triple of subsets to the set of elements ...
- 60.2k
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Structure and representation of a non-homogeneous quadratic algebra
Let $m$ be a positive integer and $s$ be a fixed $m\times m$ matrix. Let $U$ be the unital associative algebra (over $\mathbb{C}$) generated by $\{u^i_{j}\}_{1\leq i,j\leq m}$ quotient over the ...
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4
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2
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A subalgebra of a Frobenius algebra that is not again a Frobenius algebra?
A Frobenius algebra is a vector space that is both an algebra and a coalgebra in a compatible way. (See here for a precise definition.) I guess that a subalgebra of a Frobenius algebra is not again a ...
4
votes
1
answer
247
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Wedderburn decomposition of special linear groups
$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...
- 249
5
votes
0
answers
174
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Bar constructions and pushouts
Suppose that $\mathsf S$ is a span of associative algebras (or, more generally, if you'd like, any type of object admitting a bar-cobar formalism) and let $A$ be its pushout.
Is there any hope of ...
- 1,534
2
votes
1
answer
151
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Reference for lattices as algebraic structures
I want to study lattices as a structure related to ring theory. I am familiar with lattices as a beginner but I want to go further and know their connections to ring theory. Do you know a book which ...
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6
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Is the monoid of all cancellative finitely generated commutative monoids cancellative?
$\DeclareMathOperator\Mon{Mon}\DeclareMathOperator\Grp{Grp}$Let $\Mon'$ be the set of isomorphism classes of (small) commutative, unital, cancellative ($a + t = b + t$ implies $a = b$) monoids. It is ...
- 1,084
5
votes
1
answer
315
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Simple component that is not a two-sided ideal
Suppose $R$ is a semisimple ring and if $L$ is a minimal left ideal. Let $B$ be the direct sum of all minimal left ideals isomorphic to $L$ ($B$ is called a simple component corresponding to $L$). It ...
- 249
6
votes
0
answers
243
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Cardinality of a polynomial image $\pmod{p^n}$
Given $P(x) \in \mathbb{Z}[x]$ a polynomial, $n$ a positive integer and $p$ a prime, there is a result that relates $|\text{Im } P \pmod{p^n}|$ with $|\text{Im } P \pmod{p^{n+1}}|$ perhaps in terms of ...
- 61
9
votes
1
answer
244
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Linear subspaces of $\mathrm{GL}_n(\mathbb{R})$ whose inverses are also linear subspaces
$\DeclareMathOperator\GL{GL}$We will call a subset $S \subset \GL_n(\mathbb{R})$ a linear subspace if it is of the form $S = S'\cap \GL_n(\mathbb{R})$ for some $S'\subset M_n(\mathbb{R})$ which is a ...
- 2,053
3
votes
1
answer
84
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Topology of the Malcev-Neumann group ring
For a ring $R$ and a group $G$ the group ring $R[G]$ consist of maps from $G$ to $R$ with finite support.
It was shown that if the group is fully ordered them this ring can be embedded in a division ...
user491484
2
votes
0
answers
66
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Cross-ratio for projective lines over division rings
If one considers a projective line over a field $k$, then the cross-ratio $(w,x;y,z)$ is a well-known geometric tool.
But what if $k$ is not commutative, that is, if $k$ is a division ring ?
Is there ...
- 4,353
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123
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Example of a brick-infinite, tame, triangular algebra of global dimension$\geq 3$
I'm trying to compute some examples and I'm unable to come up with a following example:
What is(are) the example(s) of an acyclic quiver $Q$ with relations such that the 2-Kronecker quiver is NOT a ...
- 767
4
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0
answers
103
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Regular coherence of tensor algebras
Under what conditions on a bimodule $M$ over a noetherian commutative ring $R$ is the tensor algebra $T=T_R(M)$ regular coherent? A theorem of Gersten says this is true for a free bimodule $M$. If $M$ ...
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2
votes
1
answer
86
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Primitive group rings and endomorphism rings
It is known that, for any group $G$, there exists a group $H$ containing $G$ such that the group ring $F[H]$ for some field $F$ is primitive, see Formanek, Edward; Snider, Robert L., Primitive group ...
user491484
1
vote
0
answers
57
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 &...
- 359
0
votes
0
answers
49
views
Complemented subalgebra in a free Lie ring
A Lie ring is a triple $(G,+, [\ ,\ ]),$ where $(G,+)$ is an abelian group and $ [\ ,\ ]$ is a bilinear map satisfying
$[x,x]=0$
$[\ ,\ ]$ is bilinear
$[[x,y],z]+[[y,z],x]+[[z.x],y]=0,\ \forall\ x,...
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2
votes
1
answer
86
views
Is it true that any semisimple Jordan algebra has the unit element?
I found this theorem in Minnesota notes of Koecher [Thm. 9 p.70]:
Thm. Any semisimple Jordan algebra has a unit element.
During the text Koecher do not state that the Jordan algebra has to be finite ...
- 295
3
votes
1
answer
322
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A few reference questions about the Baker–Campbell–Hausdorff formula
I'm looking at the article Baker–Campbell–Hausdorff formula - Wikipedia and I have a few questions.
Under the "Special cases" section, there is a notation $\DeclareMathOperator{\ad}{ad}$
$$ \...
- 217
7
votes
1
answer
424
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Which monoids have a faithful irreducible representation?
Let $*$ be a binary operation on a set $M$, with an identity element $e\in M$.
A monoid representation of $(M,*,e)$ is a map $\delta:M\to (S\to S)$ for some set $S$, such that $\delta(e)=\mathrm{id}_S$...
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0
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77
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A question to the Wedderburn-Mal’cev decomposition
Excuse me, I saw the result on the Wedderburn-Mal’cev decomposition of unital compact rings which M.I. Ursul and A. Tripe introduced in the attached file. However, I cannot contact them because ...
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0
answers
29
views
Proving equivalence of tropical polynomials
I am new to the world of tropical mathematics. I am wondering if there is an algorithm to prove the equivalence of two tropical polynomials (in the plus-min semiring let's say), say over multivaribles?...
- 11
7
votes
1
answer
429
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Center of a monoid ring
According to the Wikipedia page the center of a group ring $R[G]$ is the set:
$$
\{ p | \forall g,\, h \in G.\, p(g) = p(hgh^{-1}) \}
$$
i.e. class functions which do not distinguish elements of the ...
user491484
2
votes
1
answer
108
views
Algebras determined by their globals
If $A= (A, f_1, f_2, ...f_n)$ is an algebra, then its global (sometimes referred to as complex algebras) $\mathcal{U}(A)$ is defined on the power set $\wp(A)$ in the usual way.
It is known that $\...
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3
votes
1
answer
133
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Ideals of an ordered ring
Suppose $R$ is a strictly ordered (non-commutative) ring, in particular $ab > 0$ for any $a,\, b > 0$, that is also discrete in that there are no elements between $0$ and $1$.
Now consider a two-...
user491484
2
votes
0
answers
154
views
Embedding a monoid into a group via its monoid ring
Suppose I have a monoid $(M,\, \cdot,\, e)$ equipped with a monoid homomorphism $\textrm{length} : M \rightarrow \mathbb{N}_+$ into the monoid of natural numbers under addition where $e$ is the only ...
user491484
10
votes
5
answers
644
views
Is there a $3$-commutative algebra?
Let $k$ be a field of characteristic $0$. If $m\ge2$, I denote $P_m$ the standard polynomial in $m$ non-commutating indeterminates:
$$P_m(X_1,\dotsc,X_m)=\sum_{\sigma\in\mathfrak S_m}\epsilon(\sigma)...
- 51.6k
1
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0
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Abelianization of the group of invertible elements in a finite local ring
Let $R$ be a finite local $\mathbb{F}_q$-algebra. Assume that $R\cong R^*$ as left $R$-modules. Are there any known results about the abelianization $(R^{\times})_{\mathrm{ab}}$?
(We can factor $R$ be ...
- 4,979
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1
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100
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Do you know of any indecomposable ring that has no isolated elements and is neither reversible, nor integral, nor nilpotent, nor unitary?
Let $R$ be a non commutative ring. We will say that an element of $R$ is isolated if it is zero divisor and nothing nonzero annihilates it at the same time on both sides.
Note that there are many ...
7
votes
0
answers
293
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A minimal semigroup generating subset of the additive reals
I asked this on MSE, but I was told to ask it here because it is a difficult question. Consider the additive magma of the real numbers, $(\mathbb{R};+)$. Does there exist a subset $S$ of the reals ...
- 2,063
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0
answers
111
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Prove that $B$ is a directing module
Let $A\cong\mathbb{K}Q/I$ be a finite dimensional, associative, basic $\mathbb{K}$ algebra, where $\mathbb{K}$ is algebraically closed and $Q$ is a finite Gabriel quiver on $n$ vertices and $I\...
- 767
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votes
2
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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 ...
- 2,336
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0
answers
133
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On skew monoid rings and skew ordered series rings
To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given ...
- 9,636
3
votes
1
answer
106
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Reference request for equivalent formulations of being absolutely indecomposable
I would like to ask the following question.
I am searching for a reference for the following statement:
Suppose $k$ is a perfect field. Let $A$ be a (symmetric) $k$-algebra and let $M$ be a finitely ...
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2
votes
0
answers
87
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Non-existence of idempotent via evaluation of higher order cocycle on a tuple of idempotents
The Kaplansky conjecture and Kaplanski-Kadison conjecture are classical conjectures about non existence of non-trivial idempotents in group algebra $\mathbb{C}\Gamma $ or the reduced group algebra $...
- 312
2
votes
1
answer
182
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On the equality $\{f\in C(X), f|_A=0\}+\{f\in C(X), f|_B=0\}=\{f\in C(X), f|_{A\cap B}=0\}$
Let $X$ be a Tychonoff space, let $A,B\subset X$ be closed. Let $J_A$ be the set of all continuous on $X$ real-valued functions which vanish on $A$.
For which $X$'s is it true that $J_A+J_B=J_{A\cap ...
- 5,385
5
votes
1
answer
320
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Injective modules
Let $A$ be a finite dimensional $k$-algebra and let $I$ be an injective module。My question is whether $I$ is a direct sum of finite-dimensional injective modules。
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7
votes
1
answer
252
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Properties of a filtered algebra that can be concluded from properties of its associated graded algebra
Let $F$ be a filtered algebra and let $G$ be its associated graded algebra. Some examples of properties of $F$ that can be concluded from properties of $G$:
(A) The dimension of $F$ is equal to the ...
- 1,144
6
votes
0
answers
96
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Automorphisms of algebraic Clifford algebra of a Hilbert space
Let $H$ be a real separable, infinite-dimensional Hilbert space and let
$$\mathrm{Cl}(H) = \mathcal{T}(H_{\mathbb{C}}) / \{v\otimes w + w\otimes w - 2\langle v, w\rangle \cdot \mathbf{1} ~|~ v, w \in ...
- 9,591
3
votes
0
answers
126
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the growth rate of poly-$\mathbb{Z}$ group
I am interested in the growth rate of the poly-$\mathbb{Z}$ group. Let $G$ be a poly-$\mathbb{Z}$ group, i.e $$G =(\dots((\mathbb{Z} \rtimes_{\phi_1} \mathbb{Z})\rtimes_{\phi_2} \mathbb{Z}) \rtimes_{\...
- 763
3
votes
1
answer
324
views
Graded global dimension of a graded algebra
Let $k$ be an algebraically closed field of characteristic $0$.
Let $A := k \langle x,x^{-1},y \rangle /(xy-qyx, x^{d_1}-ay^{d_2})$, where deg$(x)>0$, deg$(y)>0$, $q,a \in k^*$ and $d_1\text{deg}...
- 639
4
votes
1
answer
159
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