Questions tagged [noncommutative-algebra]
For questions about rings which are not necessarily commutative and modules over such rings.
1,543 questions
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Involution of second kind in semisimple algebras
On page 20 of the book of Involutions, I have read something that I do not understand. Let $B$ be a simple $F$-algebra with centre $K$ and an involution of the second kind $\tau$.
The author said: &...
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Evaluation of all noncommutative polynomials at a point
Let $\mathcal{P}$ be the set of all noncommutative polynomials in $n$ free variables over the field $\mathbb{R}$. Let $X_1,X_2,\dots,X_n \in M_k(\mathbb{R})$ be fixed. Is there any description of the ...
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Smallest subalgebra which commutant is trivial.
Conventions
Let $k$ be a field, $M_n(k)$ be the unitary algebra of square matrices and $T_n(k)$ its sub-algebra (also unitary) of upper-triangular matrices and $N_n(k)$ its sub-vector space of ...
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Is a center of a semisimple ring also semisimple?
I have the following question:
Show that the center of a semisimple unital ring is a product of a finite number of fields.
At first, I thought of showing that the center of a semisimple field is ...
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Existence of self adjoint idempotents in simple algebras
Let $(A,*)$ be a finite-dimensional simple $K$-algebra with involution. Does $(A,*)$ have a self-adjoint primitive idempotent?
By Wedderburn theorem, $A \cong Mat_n(D)$ for some division $K$-algebra $...
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Quaternion Algebra splits
I want to show that $(-1,b)_{\mathbb{F}}$ splits if and only if $b$ is a sum of two squares in $\mathbb{F}$. For this I want to use the fact that $(a,b)_{\mathbb{F}}$ splits if and only if $ax^2 + by^...
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Semiring with a unique left ideal
For the purpose of this post, a semiring is an algebraic structure satisfying the axioms of a (unital) ring except the existence of additive inverses.
Recall the equivalent characterizations of a ...
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Graded modules that are sum of their finite dimensional submodules.
Let $A$ and $B$ be $\mathbb{N}$-graded $k$-algebras such that $dim_k A_i < \infty$ and $dim_k B_i < \infty$ for all $i$. Let $\phi : A \rightarrow B$ be a homomorphism of graded $k$-algebras. ...
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Morita equivalent rings that is not a matrix algebra over the other
I know that the classical example of Morita equivalent rings $R$ and $M_n(R)$, for $n\ge 1$. I want to find some more examples of rings which are Morita equivalent but are not examples of matrix ...
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Give an example of a noncommutative domain in which non-trivial two-sided ideals are principal (cyclic).
A domain $R$ is called the principal left (resp. right) ideal domain if all left (resp. right) ideals of $R$ are cyclic (principal). There are some rings that are a principal left ideal domain but not ...
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Zero as sum of four squares in a field
I have the following question in regards to non-commutative algebra:
Let $F$ be a field s.t. $char(F) \neq 2$. If $0$ can be presented as the sum of four squares of elements in $F$, not all zero, ...
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What is this finite dimensional algebra?
Fix a field $k$. Consider the (non-commutative, associative) $k$-algebra $A$ with generators $x$, $y$ subject to the relations
\begin{align*}
x^2&=x\\
y^2&=y\\
x-xy-yx+y&=1
\end{align*}
...
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Small submodules and projective covers
Let $M_R$ be a module. We say that a submodule $N\subset M$ is small in $M$ (written $N \ll M$) if there is no proper submodule $V$ of $M$ with $M=U+V$ (equivalently, if $M=U+V$, then $V=M$). We say ...
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multiplication of non-commutative exponent
my result is diffrent from the equation (24) of DOI:10.1063/1.1664490 of a minus.
In short words: When using BCH formula the element A is "-b'H" which leads to the sin/sinh(-b'H). And this ...
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Can a non-commutative $R$ algebra $S$ be isomorphic as a module to direct product of copies of $R$?
Let $R$ be a commutative ring and $S$ be a $R$-algebra such that $S$ is non-commtative.
Is $S \cong R^n,~ n \geq 1$ possible?
Since a direct product or direct sum of commutative ring is also a ...
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Are $*$-Subalgebras Ever Dense in Their Double Commutant in Other Topologies?
We know $A$ a $*$-subalgebra of $B(H)$ is strongly dense in its double commutant $A''$. Are there particular conditions on $A$ or $H$ which allow us to strengthen this to one of the following:
$A$ is ...
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Real *-Subalgebras of the Real Analog of $B(H)$ and their Strong Closures
We know for $A$ a $*$-subalgebra of $B(H),$ $A$ is strongly dense in its double commutant $A''.$ Consider the following situation: $H$ is separable, $(e_n)$ an orthonormal basis on it. Let $B(H)_\...
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Every simple module isomorphic to a minimal left ideal when the algebra is Artinian
Let $A$ be an Artinian algebra. Is every simple $A$-module $M$ isomorphic to a minimal left ideal of $A$?
My work:
If there exists a minimal left ideal $N$ such that $NM=M$, then there is a nonzero ...
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multiplication of exponential of non-commutative generator
Edit(2024.10.24): sorry for post a bad. and i dont know if allowed to post again. Edited again. Also thanks to previous answer by @Roland F
Background: DOI:10.1063/1.1664490
$H,P_i,K_i,J_i$ is ...
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Every simple module is isomorphic to a minimal left ideal [duplicate]
Let $A$ be a simple Artinian algebra. I want to prove every simple $A$-module is isomorphic to a minimal left ideal $M$ of $A$.
I do it as follows:
$MA=\sum_aMa$ is a nonzero ideal, so $MA=A$. The sum ...
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About $\mathbf{C}\otimes_\mathbf{R}\mathbf{H}\cong\mathrm{M}_2(\mathbf{C})$ [duplicate]
Let $\mathbf{H}$ be the Hamiltonian quaternions.
I want to prove that $\mathbf{C}\otimes_\mathbf{R}\mathbf{H}\cong\mathrm{M}_2(\mathbf{C})$, but I have no idea on constructing the isomorphism. Is ...
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Effect of an element of group algebra in the kernel of module endomorphisms
Let $G$ be a finite group, $K$ be field, and $M$ be a $KG$ module. The $KG$-module structure of $\text{End}_K(M)$ is given by $$g\cdot \tau(x)=g\tau(g^{-1}x);\ \forall x \in M,$$
for every $\tau \in \...
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Block decomposition of equivariant maps using Wedderburn-Artin theorem
Let $K$ be a field, $A$ an Artinian simple $K$-algebra with minimal left ideal $M$.
We can view $M$ as a simple $A$-module, so $D^{op}:=\text{End}_A(M)$ is a $K$-division algebra by Schur's lemma. ...
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Order of an element in symmetric group
In $S_3$, order of $(1 2 3)$ is $3$ and order of $(1 3 2)$ is also $3$, $e = (1 2 3) \circ ( 1 3 2)$, order of $e$ is $1$. So we got $\text{odd} \times \text{odd} = \text{odd}$.
How it is possible?
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Finding the multiplicative inverse of $a^{-1}+b^{-1}$ in a non-commutative ring R, given that $a,b,$ and $a+b$ are invertible elements in R. [duplicate]
Here is the full question: Let R be a non-commutative ring and suppose $a, b,$ and $a+b$ are all invertible elements in R. Show that $a^{-1}+b^{-1}$ is invertible and find its multiplicative inverse.
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Wedderburn-Artin theorem and primitive idempotents
Let $A$ be a simple Artinian $K$-algebra. This algebra contains a minimal left ideal $M$ which can be seen as a simple left $A$-module. By Schur's lemma, $D^{op}:=\text{End}_A(M)$ is a $K$-division ...
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Rank of a submodules over non-commutative domains
I'm looking for some insight to a question on modules defined over non-commuative rings.
Let $M$ be a module of finite rank over some ring $R$, with $N\leq M$ a submodule.
If $R$ is commutative, it ...
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Reflexive modules over finite extension ring
Let $R$ be a commutative normal domain, and $\Lambda$ be a module-finite $R$-algebra. Assume $\Lambda$ is reflexive as an $R$-module. My question is simple:
Question. If a finitely generated right $\...
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Prove that the (coordinate ring of) quantum plane is Noetherian.
The quantum plane is defined as
$$
k_q\{x,y\}:=k\{x,y\}/(xy-qyx),
$$
that is, the free (noncommutative, unital & associative) $k-$algebra where we set $xy=qyx$, where $0\neq q \in k$, $k$ is a ...
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On endomorphism algebra of minimal left ideals
Let $A$ be a simple $K$-algebra with minimal left ideal $L$. By Schur's lemma $D:=\text{End}_A(L)$ is a centrally simple $K$-division algebra. $D$ acts as a ring of right operators of $L$, endowing $L$...
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Defining the universal twisting morphism of the bar construction
I'm currently studying bar-cobar adjunction in the simplest case of algebras and coalgebras. I'm stuck in understanding of universal twisting morphism. $\pi: BA \to A$, more explicitly - $\pi: BA = ...
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Noncommutative analogues of integration
In a paper I’m writing, I take the partial sums of sequences of elements from a monoid. Much of the paper has analogous theorems for continuous cases using integration. In particular, one of the ...
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trace in group algebras
Let $KG$ be the group algebra of $G$ over $K$. For $a \in KG$, we usually define $tr(a)$ as the trace of left multiplication $a: x \mapsto ax$ for $x \in KG$. In Lam's book on noncommutative rings, I ...
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Real division algebras with involution
Let $W$ be an irreducible $\mathbb{R}$-representation of finite group $G$. Schur's lemma implies that $\text{End}_{\mathbb{R}G}(W)$ is a division ring. Since its centre is $\mathbb{R}$, $\text{End}_{\...
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Minimum extension over Q to get the quintics solvable
I noticed that when we extend $\mathbb{Q}$ to the Ring $M_5(\mathbb{Q})$, the ring of $5 \times 5$ matrices, we get that the quintics over $\mathbb{Q}$ become solvable. Namely, by the companion matrix ...
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Some examples of noncommutative nil ideal
An ideal $I$ is said to be nil ideal if each of its elements is nilpotent. For example $\lbrace0,2,4,6\rbrace$ is a nil subring of $\mathbb{Z_8}$ but it is commutative, any example of non commutative?
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Involution and inner product in semisimple group algebras
Let $\mathbb{F}$ be a field with characteristic zero and $G$ be a finite group. The group algebra $\mathbb{F}G$ is semisimple, and so
$$\mathbb{F}G=\mathbb{F}Ge_1\times\cdots \times \mathbb{F}Ge_k,$$
...
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Does $R/I\cong R/J$ as $R$-modules imply $I=J$ when $R$ is not commutative?
Let $R$ be a commutative ring, and $I$ and $J$ ideals of $R$. Equip $R/I$ and $R/J$ with their natural $R$-module structures, then if $R/I\cong R/J$ as $R$-modules we have that $I=J$. Indeed, if $\phi:...
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If each of the coefficient of $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor, then $f(x)$ is a zero divisor
Let $R$ be a ring. Suppose each of the coefficients $a_i$ of $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor. Is it true that $f(x)$ is a zero divisor?
My attempt: Since $a_i$ is a zero divisor, it ...
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Product of rings are Morita equivalent implies the rings are Morita equivalent.
Let $A$ and $B$ be two rings. Then we say $A$ is Morita equivalent to $B$, denoted as $A\sim B$, if the category of left $A$-modules is equivalent to the category of left $B$-modules.
Let $A^{n}:=A\...
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For every upper triangular matrix $A$ there exists an upper triangular matrix $B$ such that $AB$ is idempotent
Suppose $A=\begin{pmatrix}
a_{11} &a_{12}& \cdots & a_{1n} \\
0 & a_{22}&\cdots &a_{2n} \\
.&.&\cdots&.\\
0&0&\cdots&a_{nn}\end{pmatrix}$ is ...
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How to define the index of a Fredholm operator between Hilbert $ A $ - modules?
Let $A$ be a $C^*$ - algebra. ( unital or non unital )
In their book : Classifying space for proper actions and K - theory of group $C^*$-algebras, pages, $10$ and $11$, Connes, Baum, and, Higson say ...
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Is it decidable if noncommutative polynomial ideal is homogeneous?
Hi everyone and thanks in advance already for any help!
I am currently wondering about the following problem:
Given $f_1, \dots,f_r \in K\langle X\rangle$, where $K$ is a field, $X$ is a finite set of ...
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Polynomial ring $R[x]$ is hypercentral if and only if $R$ is hypercentral ring
Let $R$ be a commutative ring then $R[x]$ is commutative ring, it is obvious.
Now define hypercenter as $T(R)=\lbrace a\in R | ab^n=b^na ,n=n(b,a)\ge 1, \forall b\in R\rbrace$
If all elements of $R$ ...
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Is it true that power series ring is isomorphic to infinite direct product of ring
I am wondering whether it is true that $R[[x]]$ $\cong$ $\Pi_{\alpha \geq 0} R_\alpha$.
To define an isomorphism between them we can choose $f(a_0+a_1x+a_2x^2+...+a_nx^n+...)=(a_0,a_1,...,a_n,...).$
...
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Idempotent Armendariz rings
Is there a mistake in this paper or I am understanding it incorrectly? They defined an idempotent Armendariz ring.
Idempotent Armendariz ring: A ring $R$ is said to be idempotent Armendariz (id-...
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Equivalent properties of regular (Von Neumann) rings.
A ring $R$ is called regular (in the sense of Von Neumann) if for every $r \in R$ there is an element $r’ \in R$ such that $rr’r=r$. Then the following conditions are equivalent:
(1) $R$ is regular.
(...
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If $Q_q(T)$ is hyponormal then prove that $Q_q(T)^{-1}$ is hyponormal!
If $Q_q(T)$ is hyponormal then prove that $Q_q(T)^{-1}$ is hyponormal,
where $Q_q(T)=T^2-2Re(q)T+\vert q\vert^2$ ,where $q$ is a quaternion.
we are given $Q_q$ is hyponormal so $Q_q(T)^*Q_q(T)\ge Q_q(...
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Tower of module-endomorphism rings
Let $R$ be a ring with nonzero left ideal $A$. Define $E_1=\text{End}({}_RA)$ viewed as a ring of right operators on $A$ and $E_2=\text{End}(A_{E_1})$ viewed as a ring of left operators on $A$. ...
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Double Centralizer Property in simple ring without identity
In Lam's book, $\textit{A first course in noncommutative rings}$, Theorem 3.11 states:
Let $R$ be a simple ring, and $A$ be a nonzero left ideal. Let
$D = End({}_RA)$ (viewed as a ring of right ...
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