# Questions tagged [noncommutative-rings]

Questions about rings that are not necessarily commutative.

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### Cohn localization examples

I'm working on my master's thesis, part of which involves an exposition on Cohn localization. (nlab discussion) In Free ideal rings and localization in general rings, Sec 7.4, Cohn gives a ...
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### Rings $R$ such that every [regular] square matrix with entries in $R$ is equivalent to an upper triangular matrix

Let $\text{M}_n(R)$ be the ring of $n$-by-$n$ matrices with entries in a commutative unital ring $R$. Theorem III in C.R. Yohe, Triangular and Diagonal Forms for Matrices over Commutative ...
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### Existence of nontrivial transfinite divisibility in $R$-modules

Let $R$ be a unital, possibly noncommutative ring and $s \in R$. For a right $R$-module $M$, define $Ms = \{ms \mid m \in M\}$; this is an additive subgroup of $M$, which is a module over the ...
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### Categories of modules generated under coproducts by a small set?

Question 1: For which rings $R$ does there exist a small set $S \subseteq Mod_R$ such that every module $M \in Mod_R$ is a direct sum of modules in $S$? Equivalenty, for which rings $R$ does there ...
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### Rings whose finitely-generated modules are co-hopfian

Let $A$ be a unital, possibly noncommutative ring. Dischinger showed  that the following are equivalent: For every $a \in A$, there exists $n \in \mathbb N$ such that $a^n A = a^{n+1} A$; For ...
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### Classification of finitely generated modules over non-commutative rings

Let $\Lambda$ be a commutative integral ring with an automorphism $\sigma$ (I have in mind $\mathbb Z_p[[t]]$ and $\sigma(t) = (1+t)^\alpha - 1$ with $\alpha \in \Lambda^\times$) and $R = \Lambda\{F\}$...
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### Linear algebra over non-commutative semirings

I'm reading up on linear algebra over semirings, and I'm wondering why people seem to stop short of showing an equivalence between linear transformations between free modules and matrices. It seems ...
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### Is a specific endomorphism of $A_1$ an automorphism?

Let $k$ be a field of characteristic zero, and let $A_1(k)$ be the first Weyl algebra, namely, the associative non-commutative $k$-algebra generated by $x$ and $y$ subject to the relation $yx-xy=1$. ...
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### Partially commutative elements in powers of augmentation ideal

Let $\vartheta$ a relation of parcial commutation over a set $X,$ and consider the respective free parcially commutative group $F(X, \vartheta).$ Let $K[F(X, \vartheta)]$ the parcially commutative ...
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### Noetherian ring with a “strange” idempotent ideal

Do you know a left-noetherian ring $R$ with a two-sided ideal $I$ such that: $I=I.I$; $I$ is not projective as a left $R$-module (and better, the tensor product over $R$ of $I$ with itself is not a ...
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### Relative Dickson (trace) criterion for Jacobson radical?

In the following, all algebras are associative and unital. Let $J\left(A\right)$ denote the Jacobson radical of an arbitrary algebra $A$. Recall that this is defined as the set of all $a \in A$ such ...
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### Field of definition of a finite dimensional division algebra and how to reduce it

Let F be a field, and E/F an infinite algebraic extension. Let D be a finite dimensional division algebra over E (meaning its center is also E). Is it possible to somehow gow down to a finite ...
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### What are all pairs $(R,M)$ of a ring $R$ and a two-sided $R$-module $M$ such that all endomorphisms of $M$ are scalar multiples of $\text{id}_M$?

I was playing with some endomorphism rings and got curious whether there is a classification of all two-sided (not necessarily unitary on any side) modules $M$ over a (not necessarily unital) ring $R$ ...
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### Possible values of symmetric functions evaluated on quaternions

Let $i,j,k$ the units of quaternions, in particular $i^2=j^2=k^2=-1, ijk=-1$. We will use non commutative variables $x,y,z$. Define $sym_{a,b,c}$ to be the polynomial made of the sum of monomials ...
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### Infinite Non Abelian Extensions Unramified Outside p

Let $K$ be a number field and $p$ be a fixed odd prime. Suppose $\mathfrak{p}\mid p$ is the only prime prime above $p$ in $K$, and that $p$ does not divide the class number of $K$ (I am okay with ...
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### 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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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, $\... 0answers 107 views ### Localizations of group algebras of free groups$\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$for the augmentation, and for$\...
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 ...