# Questions tagged [noncommutative-rings]

Questions about rings that are not necessarily commutative.

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### If the Grothendieck ring of a semiring on a free commutative monoid is unital, is the original semiring unital?

Suppose $S$ is an associative semiring whose underling commutative monoid is free (in particular, cancellative) and that its Grothendieck ring $G(S)$ is a unital ring. Can we conclude that $S$ must be ...
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### If, in a unit-regular ring, the right annihilator of $a$ equals the right annihilator of $b$, then $aR = bR$?

Recall that a (unital) ring $R$ is von Neumann regular (VNR) if, for each $x \in R$, there exists $y \in R$ such that $x = xyx$; and unit-regular if such an element $y$ can be taken to be a unit. ...
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### Non-singular rings which are Rickart

A ring $R$ is said to be a right Rickart ring if the right annihilator of any element in $R$ is of the form $eR$ for some idempotent $e \in R$. It turns out that a ring $R$ is right Rickart iff every ...
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### Example of an associative unital ring R with stable range 1 and Jac(R)=0 that is not an exchange ring

Rings are supposed to be associative and unital, but not necessarily commutative. Some definitions: (Bass) A ring $R$ is said to have stable range $1$ if for all $a,b \in R$, whenever $Ra+Rb=R$, ...
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### Is Hilbert basis theorem true for positive graded ring?

Let $R=\oplus_{I\geq 0}R_i$ be a positive graded ring(maybe not commutative), where $R_0$ is a commutative Noetherian ring. If $R$ is finite generated $R_0$-algebra, is $R$ Noetherian? In here, Is ...
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### Non-commutative projective lines

There have been many approaches to the notion of projective line: combinatorial approaches (e.g. as certain permutation groups, such as $\mathrm{PGL}_2(k)$ in its natural action on $\mathbb{P}^1(k)$, ...
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### Constructing a centrally primitive idempotent in the group algebra of the symmetric group

Consider the group algebra of the symmetric group $\mathbb{C} S_k$. Given some Young tableau $T$ of shape $\lambda$, let $a_{\lambda,T}$ and $b_{\lambda,T}$ be the row symmetrizer and column ...
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### Ring epimorphisms and finiteness assumptions

Let $f : A\to B$ be a ring epimorphism. It is well-known that, under the extra assumption that $A$ and $B$ are commutative, then $f$ makes $B$ a finitely-generated $A$-module implies that $f$ must be ...
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### Irreducible skew polynomials over an algebraically closed field

Let $\mathbb{F}$ be a field, and denote with $\mathbb{F}[t,\sigma]$ the skew-polynomial ring, where $\sigma$ is an automorphim of $\mathbb{F}$. Recall that the multiplication of this ring is defined ...
Let $R$ be a (commutative) ring (with identity). A nonzero idempotent $e\in R$ is called primitive idempotent, whenever it has no decomposition into $a+b$ where $a$ and $b$ are nonzero orthogonal ($... 0answers 86 views ### Is a specific endomorphism of$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$. ... 0answers 40 views ### Partially commutative elements in powers of augmentation ideal Let$\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)]$the parcially commutative ... 1answer 206 views ### Noetherian ring with a “strange” idempotent ideal Do you know a left-noetherian ring$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$with itself is not a ... 1answer 156 views ### Relative Dickson (trace) criterion for Jacobson radical? In the following, all algebras are associative and unital. Let$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$such ... 1answer 120 views ### Field of definition of a finite dimensional division algebra and how to reduce it Let F be a field, and E/F an infinite algebraic extension. Let D be a finite dimensional division algebra over E (meaning its center is also E). Is it possible to somehow gow down to a finite ... 0answers 89 views ### What are all pairs$(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$... 1answer 296 views ### Possible values of symmetric functions evaluated on quaternions$\DeclareMathOperator\sym{sym}$Let$i$,$j$,$k$be the units of quaternions, in particular$i^2=j^2=k^2=-1$,$ijk=-1$. We will use non commutative variables$x$,$y$,$z$. Define$\sym_{a,b,c}$to be ... 0answers 51 views ### Infinite Non Abelian Extensions Unramified Outside p Let$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$(I am okay with ... 0answers 326 views ### Slightly noncommutative Nakayama's lemma? Nakayama's lemma asserts the following. If$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$and$...
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 $R$ ...