Questions tagged [noncommutative-rings]
Questions about rings that are not necessarily commutative.
5
votes
0answers
160 views
Slightly noncommutative Nakayama's lemma?
Nakayama's lemma asserts the following. If $R $ is a commutative ring with an element $s$, and $M$ is a finitely generated module such that $sM = M$, then there exists $r \in R$ such that $rM =0$ and $...
1
vote
0answers
42 views
Localizing prime ideals over Noetherian rings
Let $R$ be a prime Noetherian ring which is not necessarily commutative. Consider the two natural ways to "extend" $R$: $R[x]$ and $M_n(R)$, polynomials over $R$ and $n$ by $n$ matrices over $R$ ...
8
votes
0answers
169 views
Rings that fail to satisfy the strong rank condition
In T.Y. Lam's book Lectures on Modules and Rings, a ring $R$ is said to satisfy the strong rank condition if, for every natural number $n$, there is no right $R$-module monomorphism $R^{n+1}\to R^n$. ...
1
vote
0answers
46 views
Single source shortest path over non-commutative finite idempotent semiring in Cartesian product
Let $G$ be a Cartesian product of two arbitrary directed weighted graphs $M$ and $N$.
The weights are from a non-commutative finite idempotent semiring.
Do there exist advanced results on the single ...
5
votes
0answers
54 views
von Neumann regular ring homomorphisms
Let us call a ring homomorphism $f\colon R\rightarrow S$ von Neuman regular if it has the property that for every left $S$-module $M$, the left $R$-module $f^*M$ is flat.
In particular, $\mathrm{id}...
5
votes
0answers
39 views
Do twisted group rings of free abelian groups admit universal fields of fractions?
Let $R$ be an associative ring with unit. Recall that an epic $R$-field is a ring epimorphism $\alpha\colon R\to D$ to a skew field/division ring $D$. An epic $R$-field $\alpha$ is called a field of ...
6
votes
0answers
165 views
Independence of characters with respect to polynomials
I came across the following property :
Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ without zero divisors,
$\mathcal{U}=\mathcal{U}(\mathfrak{g})$ be its enveloping algebra. As such, $\...
6
votes
0answers
97 views
Localizations of group algebras of free groups
$\newcommand{\QQ}{\Bbb Q}$
Let $G$ be a free group on the symbols $x_1, \dots, x_n$, with $\QQ[G]$ its rational group algebra.
Write $\varepsilon: \QQ[G] \to \QQ$ for the augmentation, and for $\...
1
vote
0answers
38 views
Relation between left projections
Let $A$ be a Baer *-ring. Let $x$ be in $A$, $L(x)$ is the left projection of $x$ that is the smallest projection with $L(x)x=x$.
Q. Let $p,q$ are projections in $A$ with $p\leq q$. For a given ...
1
vote
0answers
35 views
The statue of a sequence of finite projections
Let $A$ be a Baer $*$-ring. Let $\{p_n\}$ be a sequence of finite projections in $A$. True or false?
Suppose that there is no $N$ with $p_n=p_{n+1}$ for $n\geq N$. We have then $\inf_{1\leq n\leq ...
1
vote
0answers
34 views
something concerning finite projections
Let $A$ be a Baer *-ring. Let $x$ be an isometry (meaning $x^*x=1$ where $1$ is the unit of $A$).
Let $e$ be a finite projection in $A$ such that $ex^ne=ex^n$ for every $n\geq0$.
Q. Can we say that ...
2
votes
1answer
69 views
Strongly finite projections in $*$-rings
Let $A$ be a $*$-ring. Let us have some points:
i) We recall that a projection $p$ is a self-adjoint idempotent that is $p=p^*=p^2$.
ii) On the set of projections, we write $p\leq q$ if $pq=p$.
iii)...
6
votes
0answers
282 views
Curious anti-commutative ring
Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context?
By $\Lambda$ I mean the free anti-commutative algebra, $x_i x_j = - x_j x_i$,
either ...
6
votes
1answer
171 views
Is every (left) graded-Noetherian graded ring (left) Noetherian?
I call a $\mathbb{Z}$-graded (non-commutative, associative, unital) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) ...
3
votes
1answer
84 views
Extensions of modules of type $FP_n$
Let $R$ be a ring (not necessarily commutative, but with a unit). Recall that an $R$-module $M$ is of type $FP_n$ if $M$ has a partial projective resolution of length $n$ whose terms are all finitely ...
2
votes
0answers
138 views
Algebraic version of unilateral shift
It was confirmed that Wold-type decomposition can be extended from von Neumann algebras to Baer*-rings (see this paper). By algebraic tools the notion of unilateral shifts is successfully transmitted ...
7
votes
0answers
124 views
Torsion in a tensor product over a group ring
Let $\Gamma$ be a finitely generated dense subgroup of a pro-$p$ group $G$. Let $\mathbb Z_p$ be the ring of $p$-adic numbers. Denote by $\mathbb Z_p[[G]]$ the completed group algebra.
Is it true ...
7
votes
1answer
173 views
invertibility of matrix over free associative algebra
For a commutative ring $R$, a matrix $A \in M_n(R)$ is invertible iff $\det (A)$ is a unit in $R$. Is there a similar criterion to determine invertibility (having two-sided inverse) of a matrix over a ...
4
votes
1answer
112 views
number of indecomposable summands of an extension of two modules
I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of ...
3
votes
1answer
213 views
Looking for example of quotient of group algebra by ideal of group ring which fails to be injective
I am looking for an example of a group ring $\mathbb{Z}[G]$ of a finite group $G$ along with a lattice $I$ (in the case at hand the word 'lattice' means: a $\mathbb{Z}[G]$-submodule which is ...
2
votes
0answers
33 views
A question related to reflexive rings
Let $R$ be a ring. An endomorphism $\alpha:R\to R$ is said to be right central reflexive if for all $a,b\in R,$ $aRb=(0)\implies bR\alpha (a)\subset Z(R)$, where $Z(R)$ denotes the centre of the ring. ...
2
votes
1answer
73 views
From socle of quotients to socle of ring itself
Let $I_1, \dots , I_n$ be ideals of a ring $R$ with identity having zero intersection. Assume that for some $x\in R$, $x+I_ i$ is an element of the right socle of $R/I_ i$, for each $ i=1,\dots , n$....
1
vote
0answers
65 views
When is $R/Soc(R)$ reduced?
Let $R$ be a ring with identity. It is readily checked that when the quotient $R/S_r$ is reduced, the nilpotent elements of $R$ fall into $S_r$, where $S_r$ is the right socle of $R$. Is the converse ...
-1
votes
1answer
87 views
Nilpotent Elements and the Socle [closed]
Let $R$ be a ring with identity such that the quotient ring $R/S_r$ is abelian, i. e., all idempotents of the quotient are central. Here, $S_r$ means the right socle of the ring $R$. Do the nilpotent ...
2
votes
0answers
46 views
An Abelan quotient ring by Socle
Let $R$ be a ring with identity whose (right) socle $S$ contains its nilpotent elements. Is it necessarily true that the quotient $R/S$ is an abelian ring? ( By an abelian ring I mean a ring whose ...
6
votes
0answers
66 views
m-systems and n-systems in topological rings
Note that throughout rings have a multiplicative identity and are not necessarily commutative
Definition: Let $R$ be a ring and let $M\subseteq R$. Then, $M$ is an m-system iff for every $x,y\in ...
2
votes
1answer
51 views
Nontrivial primitivity of full matrix ring
Let $V$ be an infinite dimensional vector space, $A$ its algebra of linear endomorphisms (also known as a full matrix ring). The algebra $A$ is primitive because $V$ is a faithful simple (left) $A$-...
2
votes
2answers
85 views
Uniqueness of character for Z_+-rings
I have a question about the proof of proposition $3.3.6(3)$ in "Tensor Categories" by Etingof et al..
This part states that for $A$, transitive unital $\mathbb Z_+$-ring, there is a unique character ...
2
votes
0answers
104 views
Existence of right adjoint for a certain functor from rings to bimodules
Fix a (possibly noncommutative) ring R and consider the category $Rings_R$ of rings $S$ with a map $S \to R$. There is a extension of scalars functor to R-bimodules $Rings_R \to R-Mod-R$ which sends $...
0
votes
0answers
84 views
Koethe Conjecture and a Nil Problem of Jacobson Radical
Let $R$ be a ring with identity, and $e^2=e\in R$ such that both $eJe$ and $(1-e)J(1-e)$ are nil, where $J=J(R)$ is the Jacobson radical of $R$. When $R$ is commutative, it is easy to see that $J$ is ...
3
votes
1answer
198 views
Why does there exist a non-split sequence with the condition that $\mathrm{pd} M=\infty$?
Why does there exist a non-split sequence with the condition that $\mathrm{pd} M = \infty$?
Remarks.
I am reading
Andrzej Skowroński, Sverre O.Smalø, Dan Zacharia: On the Finiteness of the Global ...
1
vote
1answer
84 views
Maximal submodule not containing an element
[Title was: Let $M$ be a module, $N$ a submodule in $M$, conditions on $x$ and $M$ ($N\leq M$ or existence of $K\leq M$ maximal $x\not \in K N\leq K$) such that $Rx+N=M$]
Let $M$ be a left $R$ module....
4
votes
1answer
155 views
Does $R$ is dedekind-finite imply $\mathbb{M}_n(R)$ is dedekind-finite
Following Lam's notation, a ring (with identity) $R$ is called dedekind-finite if $ab=1\iff ba=1$ in $R$.
There are a lot of result about left invertible implies right invertible. But the results ...
5
votes
1answer
226 views
Where can we find polynomial's root?
Let $R$ be a ring (not commutative in general) with identity and let
$\Psi(x) = x^m-\sum_{j=0}^{m-1}\psi_jx^j$ be a monic polynomial over $R$. I want to construct a ring extension $K$ of $R$, which ...
3
votes
0answers
28 views
subisomorphic modules
An $R$-module $M$ is called Baer if for every $N\leq M_R$, ann$_{S}(N)$ is a direct summand of $S$ where $S=$ End$_{R}(M)$.
Question: Let $N$ and $K$ be Baer $R$-modules such that $N$ is isomorphic ...
4
votes
1answer
102 views
If $\{f\in R[x]\:|\:f\text{ monic}\}$ is a right denominator set, is $\{f^i\:|\:i\geq 0\}$ a right denominator set also?
Let $R$ be a right (and left) Noetherian ring and $T=R[x]$ its polynomial ring. It was shown by Stafford that the set $S=\{f\in T\:|\:f\text{ monic}\}$ is a right denominator set. So my question is, ...
2
votes
1answer
126 views
On Artinian rings
Let $R$ be a ring, and for every $R$-module $M$, suppose that we have the following condition:
If $M$ is cogenerated by any finitely generated $R$-module $N$, then $M$ embeds in a finite direct sum ...
11
votes
0answers
195 views
Finitely generated skew-fields
There is a well known theorem saying that a commutative field that is finitely generated as a ring has to be finite (Kaplansky).
Is the same true for non-commutative "fields" (usually called ...
3
votes
0answers
52 views
Reference request: Hecke agebra over non-commutative rings
I think the title sums it up quite well: Is it a useful idea to define the Iwahori-Hecke algebra over a non-commutative $k$-algebra? If so, what shape should the relations attain?
Bonus question: ...
4
votes
0answers
180 views
Algebraic-closures of division rings
In what follows, $x$ is always taken to commute with the coefficient ring. This means that for any given polynomial, you can put the coefficients to the right or the left of $x$ as you please. This ...
3
votes
1answer
104 views
When the endomorphism ring of the injective envelope of any simple module is division ring?
I have a question. I will be thank you if you give me some hints.
This is the question:
Let $R$ be a ring and for any simple R-module T, we have End(E(T)) is isomorphic to End(T) which is a division ...
4
votes
0answers
204 views
Faithfully flat descent of projectivity for non-commutative rings
I am looking for a reference for the following statement (or another one explained further below):
Let $M$ be a module over a (not necessarily commutative) ring $R$ and $R'\supset R$ a faithfully ...
6
votes
1answer
163 views
Quotients of rings with finite free additive group
Let $R$ be a ring (assumed associative and unital) whose additive group is a finitely generated abelian group. As a reduction step in a paper I'm working on, we need to know that $R$ is a quotient of ...
1
vote
0answers
56 views
A sufficient condition for finite *-rings!
Let $R$ be a unital *-ring. Assume that $R$ has finitely many projections.
Q. Can we conclude that $R$ is finite?!
$\bullet$ We say $R$ is finite if $x^*x=1_R$ implies that $xx^*=1_R$.
$\bullet$ ...
1
vote
0answers
50 views
Any link between abelian $R/J(R)$ and 2-primal condition
Let $R$ be noncommutative unital ring such that each element of the quotient $R/Soc(R_R)$ is idempotent. If the nilpotent elements of $R$ form an ideal, is it true that the idempotents of $R/J(R)$ ...
3
votes
0answers
66 views
The going-up theorem for free extensions of almost commutative rings
I would like to know whether or not the going-up property holds for some classes of finite filtered extensions of non-commutative rings.
Let $S \subseteq R$ be rings. The pair $(S,R)$ has the going-...
2
votes
0answers
89 views
A Boolean quotient ring of a prime ring
I am searching for a unital prime ring $R$ such that its right socle $Soc(R_R)$ is nonzero and proper, and such that $R/Soc(R_R)$ is a Boolean ring (i.e., all its elements are idempotent).
Thanks for ...
2
votes
1answer
49 views
Commutative inner inverse for non-unital strongly regular ring
An element $a$ in a ring $R$ is called strongly regular if $a \in a^2R$ and $a \in Ra^2$, in other words $a = a^2x$ and $a = ya^2$ for some elements $x,y \in R$. Say that $R$ is a unital ring. $a \in ...
1
vote
0answers
36 views
Does the Polarization Theorem for $A_1(k)$ has an analogue for $k[x,y]$?
There is an interesting theorem about the first Weyl algebra $A_1(k)= k \langle x,y | yx-xy= 1 \rangle$,
$k$ is a field of characteristic zero,
the Polarization Theorem, Corollary 5.5 by A. Joseph.
...
1
vote
1answer
111 views
Non-commutative Ito Formula
Does there exist a formula of the Ito lemma for matrix valued processes under matrix multiplication?
That is where
$$
\Delta X_{t+\Delta t} \neq X_{t+\Delta t} - X_t
$$
but instead
$$
\Delta X_t = ...
