# Tagged Questions

Questions about rings that are not necessarily commutative.

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### Whether a given algebra is the algebra of endomorphisms for a vector space

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...
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### Making a multivariate polynomial monic in one of its variables

I apologise in advance for the general nature of this question. Suppose we have a non-commutative ring $R$ that is relatively well-behaved as non-commutative rings go (I was thinking of $R$ being the ...
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### What are the fixed points of $\beta_j^{-n}[\alpha^n--\beta_j^{n-1}\mu_j-\beta_j^{n-2}\mu_j-…-\mu_j]$ for a fixed $j$ [closed]

Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= \beta_i x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$. ...
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### What are the fixed points of $\alpha^n-\mu_j$ for a fixed $j$?

Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$. What are ...
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### In commutativity theorems in ring theory

Suppose that $R$ is a ring such that for any $x\in R$ there exists $1<n(x)\in \mathbb{N}$ such that $x^{n(x)}-x\in Z(R)$. Prove that $R$ is commutative or if it is not commutative, then the ideal ...
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### the relation between projective and quasi-projective modules

An $R$-module $M$ is called quasi-projective if $\text{Hom}_R(M,M)\to\text{Hom}_R(M,N)$ is surjective for every surjective homomorphism $M\twoheadrightarrow N$. What are the rings $R$ for which every ...
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### Attaching an ideal whose square is zero: does this operation have a name and a notation?

I know I met the following construction somewhere, but I cannot remember where. Let $A$ be a (unital associative) ring, and let $N$ be an $A$-$A$ bimodule. On the product set $A\times N$ we define ...
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### Computing intersection of Weyl algebra ideal with certain subring

Let $D=k [x_1,\ldots, x_n, \partial_1,\ldots, \partial_n]$ be the nth Weyl algebra over the characteristic zero field $k$. Set $\theta_i=x_i\partial_i$. Let $I$ be a left ideal in $D$. Is there a ...
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### Maximal commutative subrings of the endomorphism ring of a vector space

Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...
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### A paper by Y. Morita

The corresponding bibliographical details are: Yoshihito Morita, Elementary proofs of the commutativity of rings satisfying $x^{n}=x$. Mem. Defense Acad. 18 (1978), no. 1, 1–24. Does anybody here ...
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### Free algebra and ring of quotients [closed]

I am reading of example T.Y.Lam 'A first Course in Noncommutative Rings': Let $R=\mathbb{Z}\langle x,y\rangle/(y^2,yx)$. To work with $R$, we shall confuse $x,y$ with their images in $R$. Thus,we ...
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### Ergodicity of elementary symmetric polynomials with noncommutable variables

Let $\{X_n\}$ be an ergodic sequence of random variables, $X_n:(\Omega,\mathcal{F})\to (S,\mathcal{S})$ where the target set $S$ is a matrix ring. My question is, Can the following limit be found ...
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### In a Noetherian domain, what can one say if all non-zero non-units are in all but finitely many maximal right ideals?

Let $R$ be a (not necessarily commutative) Noetherian domain. Clearly, units of $R$ are not in any maximal right ideal, and elements of $J$, the Jacobson radical, are in all maximal right ideals. All ...
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### Factorisation of twisted polynomials

Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...
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### Quaternion algebra in characteristic $p$

Given a prime number $p$, can you give me concrete examples of fields $\mathbf F$ of characteristic $p$ and quaternion algebras $\mathbb H(\mathbf F)$ over $\mathbf F$ such that $\mathbb H(\mathbf F)$ ...
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### Differentially closed fields

Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$. Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...
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### Existence of a skew field with surjective inner derivations

In my research, I've come twice now towards a skew field $K$ that satisfies the following: $$\text{for all non-central element a, the map }\quad x\mapsto ax-xa\quad\text{ is onto.}$$ I am hoping ...
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### Sums of squares via semidefinite programming for the complex free group algebra

In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and ...
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### Generation of cohomology of graded algebras

Let $A$ be an unital, associative, graded algebra over a base ring $k$. I'm happy to assume that $k$ is a field if need be, and will insist that $A$ free and of finite rank in each degree (locally ...
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### Must a finitely generated projective module over a group ring with vanishing coinvariants be trivial?

Let $G$ be a (possibly infinite) group. Let $\mathbb{Z}[G]$ be its integral group ring and let $P$ be a finitely generated projective module over $\mathbb{Z}[G]$. Suppose that the coinvariants of $P$ ...
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### How to prove that a projective module is not free?

Let $A$ be a noncommutative (perhaps $\ast$-) algebra (over $\mathbb{C}$) and let $M$ be a projective module defined via a projector $P\in M_n(A)$; i.e. $M=P(A^n)$. Furthermore, assume that all ...
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$\newcommand{\ad}{\operatorname{ad}}$As my recent post (here) did not receive any answers yet, I thought I would ask a similar question in which I'm also interested. Let $A=$ $^{k \langle x,y\rangle }... 0answers 68 views ### Torsionfree finitely generated compact Iwasawa module The following fact falls under the category of Iwasawa modules. Let$M$be a torsion free finitely generated module over the non commutative noetherian ring$\Bbb{Z}_p[[G]]$, (where$G$is a p ... 1answer 150 views ### If$R$is generated by idempotents, then$\text{Ann}(R)=0$? Let$R$be a ring (not necessarily commutative or unital) that is generated by idempotents. I'd like to know if$\text{Ann}(R)=0$must hold. Here I use$\text{Ann}(R)$to denote the set of all ... 0answers 79 views ### Counting models in first order logics without existencial quantifiers My question is about the posibility of to construct a parameter space of models in a first order theory, finitely presented, with out existencial quantifiers (parameter space in the sense of ... 0answers 44 views ### About alternating polynomials an the Rowen's notation Some definitions... Definition 1: A polynomial$f(X_1,\dots ,X_d)$is$t$-linear if the variables$X_1,\dots ,X_t,\; t\leq d$appear in all monomials of$f$and degree of$X_i,\; i=1,2,\dots ,t$on ... 0answers 77 views ### Projective dimension of ring over its center If$A$is a ring and$Z(A)$is its center then what is a sufficient condition for the projective dimension of$A$over$Z(A)$(ie:$pd_{Z(A)}(A)$) to be finite? (Assuming that$A\neq Z(A)$). 1answer 157 views ### Is the following module over a group ring necessarily infinitely generated? Suppose$\Gamma$is a (finitely presented, but this is probably irrelevant) group, and$M$is a finitely generated (EDIT: finitely presented) module over$\mathbb{Q}\Gamma$which is infinite-... 3answers 373 views ### smooth connected affine scheme over Z has good reduction almost everywhere Let$f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$be a polynomial. Assume that the variety cut out by$f$is smooth and connected (so irreducible) over$\overline{\mathbf{Q}}$. Where can I find a ... 0answers 100 views ### Examples of noncommutative Bezout domains I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ... 0answers 48 views ### extend derivations of ore domain to its quotient field I wonder whether someone knows a good reference(textbook or paper) for the following result: Any derivation of ore domain may be extended unqiuely to a derivation of its quotient field. Thanks. 1answer 180 views ### What do epimorphisms in noncommutative rings look like? The question I want to ask is inspired by this mathoverflow post about epimorphisms in the category of commutative rings. I found the seminar (by P. Samuel) referenced by David Rydh particularly ... 0answers 92 views ### Relationships between finiteness of stable rank and IBN property of rings Does any ring of finite stable rank have IBN property? Where can we find this result? 1answer 121 views ### Injective modules over noncommutative noetherian rings Let$R$be a left noetherian ring, then it is well known that Matlis proved that any injective$R$-module decomposes into a sum of indecomposable injectives, each of form$E(R/I)$for some irreducible ... 0answers 272 views ### Reconstruction of noncommutative scheme It is known that a quasi compact scheme(even quasi separated scheme)can be determined uniquely by the category of quasi coherent sheaves on it by Gabriel-Rosenberg reconstruction theorem The ... 2answers 313 views ### Maximal centralizer in full matrix ring I will be so thankful if someone can help me with the following question. Is it possible to obtain all maximal centralizers in the full matrix ring,$M_n(F)$, for an arbitrary finite field$F$? Here, ... 2answers 276 views ### Polynomial identities for mod p matrices Can there be a polynomial over the field$F_p$of$p$elements ($p$prime) in non-commuting variables$X_1,..., X_r$such that: 1)$f(A_1,...,A_r)=0$for every$n \times n$matrices$A_1,...,A_r$... 1answer 162 views ### What are the upperbounds of the Nil radical? The main radicals of a non-commutative ring (with 1) are the Sum of all nilpotent ideals$\subseteq$Prime radical$\subseteq$Nil radical$\subseteq$Jacobson radical$\subseteq$Brown-McCoy radical. ... 1answer 181 views ### Non-simple and non-unital rings with trivial centres Let$R$be an associative and non-unital ring. (Suppose that$R$is$s$-unital, i.e. for each$x\in R$there is$u,v\in R$such that$ux=xv=x$.) It is not difficult to show that if$R$is a simple ... 0answers 57 views ### Does Castelnuovo-Mumford regularity hold for this$\mathbb{C}$-algebra$?

Let $R$ be a noncommutative finitely generated $\mathbb{C}$-algebra such that its center $S$ is smooth (in commutative sense) and $R$ is finite over $S$. Is there Castelnuovo-Mumford regularity ...
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### Noncommutative computational package

I am wondering if there is a program which can do simple operations over noncommutative rings, like expand products and substitute one expression for another. To clarify, consider the following ...
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### Conceptual explanation of Strassen's trick for matrix multiplication

Algorithms for fast multiplication of polynomials and integers have well-known conceptual explanations. A good survey paper is Daniel J. Bernstein's Fast Multidigit Multiplication for Mathematicians. ...
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### A graded ring $R$ is graded-local iff $R_0$ is a local ring?

I asked this question some months ago on math.stackexchange.com: http://math.stackexchange.com/questions/126810/a-graded-ring-r-is-graded-local-iff-r-0-is-a-local-ring It would be great (for me) to ...
I've been looking for the following construction in the literature, but I've only been able to find (very) partial proofs or proofs of special cases. Let $R$ be a non-commutative ring and $S$ a ...
Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples? The question popped up ...