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 finite). Further, A is connected: it vanishes in negative degrees and is of rank 1 (generated … Read more

Book or survey on Dedekind-finite rings

I’m seeking a book or a survey providing an overview, as rich as possible, of the literature on Dedekind-finite (or von Neumann-finite) rings (let me recall that a unital ring R is Dedekind-finite if xy=1R for some x,y∈R implies yx=1R). There is something in C. Faith’s Algebra II – Ring Theory (1979) and other classical … Read more

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 doesn’t make a difference to the ring itself, but it will make a difference for the roots: a … Read more

A non-commutative, left duo ring whose only unit is the identity

Let $R$ be a ring (here, rings are always associative, unital, and non-zero). We say that $R$ is a left duo ring if $aR \subseteq Ra$ for every $a \in R$. Question. Is there a non-commutative, left duo ring whose only unit is the identity? It is perhaps worth noting that, if the only unit … Read more

Possible values of symmetric functions evaluated on quaternions

Let i, j, k be the units of quaternions, in particular i2=j2=k2=−1, ijk=−1. We will use non commutative variables x, y, z. Define syma,b,c to be the polynomial made of the sum of monomials which are all possible products of a variables x, b variables y and c variables z. For example sym2,1,0(i,j,k)=i2j+iji+ji2. Considering the … Read more

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 projective left R-module)? Answer Take any idempotent e in a finite dimensional quiver algebra KQ/L , then the … Read more

Does RR is dedekind-finite imply Mn(R)\mathbb{M}_n(R) is dedekind-finite

Following Lam’s notation, a ring (with identity) R is called dedekind-finite if ab=1⟺ba=1 in R. There are a lot of result about left invertible implies right invertible. But the results all require some finiteness property on the ring or the matrix ring. I am asking a proof or a couterexample of that that R is … Read more

What is the extended centroid of a free algebra?

For a prime ring $R$, you can define its “Martindale ring of quotients” $Q(R)$. See for example: Martindale, Wallace S. III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12, 576-584 (1969). ZBL0175.03102. The center of this ring is called the “extended centroid” of $R$. It is known that it is a field. I … Read more

Locally nilpotent operators of the Weyl algebra

$\newcommand{\ad}{\operatorname{ad}}$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 $A=$ $^{k \langle x,y\rangle }\Big/_{(yx-xy-1)}$ be the Weyl Algebra over a field $k$ of characteristic $0$. For an element $q(x,y) \in A$, we have the operator $\ad_q(h):=[q,h]=qh-hq$. Also, we say … Read more

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, if $g\in S$, then is $G=\{g^a\:|\:a\geq 0\}$ a right denominator set also? If $t\in T$ and $g^i$ then I want to … Read more