# Questions tagged [noncommutative-algebra]

For questions about rings which are not necessarily commutative and modules over such rings.

1,485 questions
Filter by
Sorted by
Tagged with
8 views

• 405
1 vote
47 views

• 405
42 views

### Is a surjective endomorphism of a finitely generated module over a *non-commutative* ring with unity necessarily an isomorphism?

As is well known, any surjective endomorphism of a finitely generated module $M$ over a commutative ring with unity $R$ must be an isomorphism. What about the non-commutative case? In other words, is ...
• 675
94 views

### Technical question about computing the left ideals of the ring $\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$

Computing the left ideals of $\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{2}$. This is the same that computing the ideals of $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}$, ...
• 1,947
122 views

### Show a ring is commutative if $r^2 = r + r$

The claim is that any ring $R$ in which for all $r \in R$ we have that $rr = r + r$, must be commutative. No assumptions are made about $R$ having multiplicative identity or being commutative. I was ...
• 73
105 views

### Compute the left ideal of the upper triangular matrix with entries in $\mathbb{Z}_{2}$. [duplicate]

An apparently similar question has been asked in Is there a nice way to classify the ideals of the ring of lower triangular matrices? But it's not the same since the those are not upper triangular ...
• 1,947
67 views

### coproduct(free product) of algebras is free iff both factors are free algebra?

Coproduct of associative algebra is defined in wiki， let us work on field so that coproduct exist generally. And for free algebra I mean semigroup ring generated by semi free groups on $n$ alphabet ...
• 389
1 vote
61 views

### Can there be a subgroup $K$ of a group $G$ such that for some $a \in G$, $aK \subseteq Ka$ but $Ka \nsubseteq aK$? [duplicate]

I have a question, in a sense, about how asymmetric left and right cosets can be when dealing with an infinite, non-normal subgroup $K$ of a (non-abelian) group $G$. Specifically, my question is ...
• 191
1 vote
46 views

### Artin-Wedderburn application - finding simple modules

I'm trying to solve the following: Let $R$ be the $\mathbb{R}$-algebra $R:=M_2(\mathbb{R})\times M_3(\mathbb{H})\times\mathbb{C}\times\mathbb{C}$. Determine how many simple (left) $R$-modules there ...
• 1,766
46 views

• 11
69 views

### Is there always an automorphism distinct from identity in a simple module?

Let $M$ be a simple module over a unital ring $R$. If $|M| \leq 2$, then $M$ has only one automorphism (the identity). I'm wondering whether for $|M| > 2$, there always exists an automorphism ...
• 1,158
54 views

### How to prove when $R$ is not a commutative ring, $M\otimes_R N$ is not a module

I know how to prove when $R$ is a commutative ring, $M\otimes_R N$ is a module, but how to prove when $R$ is non-commutative ring, we cannot give any module structure on $M\otimes_R N$? I see many ...
• 117
71 views

### Using Bergman's Diamond Lemma to prove a PBW theorem

I've found myself in a situation where I need to prove a PBW theorem for a certain quotient of an algebra $A/I$ where $A$ is a filtered algebra given by generators and relations. Furthermore $A$ has a ...
• 334
1 vote
33 views

### Two algebra structures on endomorphisms

Let $(\mathcal{M}, \otimes, \mathbb{k})$ be a symmetric closed monoidal category, which in my application is the category of $dg$-modules over some commutative ring. Let $A$ be a bialgebra/bimonoid in ...
43 views

### Lie algebra cohomology of formal non-commutative vector fields

Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
• 35
1 vote
48 views

### To what extent are submodules of $R^n$, for a ring $R$, describable by systems of linear equations?

Let $R$ be a ring and consider $R^n$ as an $R$-module. I'm wondering under what circumstances is a submodule of $R^n$ the solution set of some homogeneous system of linear equations in $n$ variables, ...
• 1,158
1 vote
29 views

### Does this property generalize to modules over noncommutative rings?

Let $M$ be a finite simple module over a ring $R$. For commutative $R$, it is easy to show that for every $r \in R$, the function $M \rightarrow M; x \mapsto rx$ is either bijective or constantly $0$. ...
• 1,158
1 vote
46 views

### Comultiplication on the tensor algebra

Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal ...
79 views

### Non commutative Non reduced ring

Let $R$ be a ring. $R$ may not be reduced and commutative. Is it true in general ring that if for any $a,b,c\in R$, we have $abc=bca=cab=0$ then it implies $acb=0$. We can prove this in case of ...
• 335
83 views

1 vote
73 views

### Finite Group's Group Algebra over a Field is a Principal Ideal Ring?

A (left) principal ideal ring (PIR for short) is a ring such that for every left-ideal $I$, there exists a $a \in R$ such that $I=Ra$. Firstly, similar to the case of PIDs, is that a ring $R$ is a PIR ...
• 358
1 vote
49 views

### An identity in the generalized quaternion algebra

The generalized quaternion algebra $D$ is defined relative to a field $F$ of characteristic $2$ with parameters $a, b \in F^\times$ by $$D = F\langle i, j \rangle/(i^2 = a, j^2 = b, ij = - ji).$$ This ...
1 vote
20 views

### Noncommutative Nakayama's Lemma for Maximal Ideal Inclusion

In "Grobner Bases and the Computation of Group Cohomology" Hypothesis 1.5 is: Let $k$ be a field of characteristic $p$. Let $\Lambda$ be a finite dimensional $k$-algebra (associative with ...
• 1,441
22 views

### A normal regular element in a connected graded ring $S$ can become central by a suitable Zhang twist

I am reading the article Noncommutative quadric surfaces (here) by S. Paul Smith and M. Van den Bergh. There is a sentence in the introduction to the article: $S$ is a not-necessarily-commutative ...
• 380
1 vote
94 views

### Derivations in a central simple algebra

In Farb's Noncommutative Algebra, the following exercise is given. Theorem 3.22 Let $R$ be a finite dimensional central simple $k$-algebra. Every $k$-linear derivation on $R$ is inner. I solved the ...
• 512
169 views

### Is $f(\operatorname{rad} A ) \subseteq \operatorname{rad} B$ when $f$ is a not surjective $K$-algebras homomorphism and $K$ is a field?

Let $K$ be a field. For a $K$-algebra $A$ take the definition of the Jacobson radical of $A$ as the intersection of all maximal left ideals of $A$. If $A$ and $B$ are two finite dimensional $K$-...
• 2,459
Let $A$ and $B$ be two matrices. I am trying to prove the following formula (and also find the conditions on $A$ and $B$ for it to work, if it is not true for any $A$ and $B$) : \underset{n=0}{\sum^...
### $f(m)=0$ for every $f\in \text{Hom}(M,S)$ then $m\in \text{rad}(M)$
Let $M,S$ be two right $A$-modules, the latter simple. Assume that for a certain $m\in M$ we have that $f(m)=0$ for every $f\in \text{Hom}(M,S)$. I want to show that then $m\in \text{rad}(M)$, where ...