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Questions tagged [noncommutative-rings]

Questions about rings that are not necessarily commutative.

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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 $...
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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 $R$ ...
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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 $R^{n+1}\to R^n$. ...
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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 ...
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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}...
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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 ...
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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, $\...
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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 $\...
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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 ...
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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 ...
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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 ...
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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)...
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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 ...
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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) ...
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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 ...
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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 ...
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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
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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
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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
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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 ...
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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
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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$....
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 $...
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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 ...
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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 ...
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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....
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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 ...
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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
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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 ...
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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
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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 ...
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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 ...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $xx^*=1_R$. $\bullet$ ...
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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 $R/J(R)$ ...
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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-...
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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 ...
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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 ...
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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. ...
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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 = ...