1-ab invertible => 1-ba invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one? Geometric series. In ...
Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group. A couple of good ...
Theo's question made me wonder if there are other "noncommutative analogs" outside of operator algebras. Some noncommutative analogs from operator algebras include: A $C^\ast$-algebra is a ...
It is well-known that the category of affine schemes is equivalent to the opposit category of commutative unital rings. So naively, one would think that the same should hold in non-commutative ...
I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task: I have two graded (generally ...
According to Corollary 1.2(3) of the paper Silver: Noncommutative Localizations and Applications. J. of Alg. 7(1964), 44-67: If $R$ is a (commutative) field and $\alpha: R \to S$ an epimorphism in ...
Definition and context: An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
I asked this question on SE a long time ago, but never received an answer: The Govorov-Lazard Theorem states that a (left) module over an unital ring is flat iff it is a direct limit of finitely ...
By inspecting the accepted answer to this question Are epimorphisms from a division ring isomorphisms ? one obtains the following necessary condition for epimorphisms: Let $R \le S$ be rings ...
I'm interested in knowing/collecting some properties of epimorphisms of rings (with identity) that are true over commutative rings but are false in the non-commutative case. Example: I learned from ...