Articles of noncommutative algebra

Non-Examples for Krull-Schmidt-Azumaya

I am looking for some rings $A$ and finitely generated $A$-modules $M$, where the conclusion of the Krull-Schmidt-Azumaya Theorem does not hold for $M$, i.e. where $M$ can be written as direct sums of indecomposable modules in multiple ways (the summands not being unique up to order and isomorphism). To be specific, I am looking […]

Correct definition of bilinear(multilinear) maps over noncommutative rings

I wonder about the following: Let $R$ be a noncommutative ring with $\alpha, \beta \in R$ such that $\alpha \beta \neq \beta \alpha$. Let $M,N,P$ be left-$R$-modules. Why does it make problems to define a map $B: M \times N \to P$ to be bilinear if $B(\lambda m,n) = \lambda B(m,n)$ and analogously for the […]

The Jacobson Radical of a Matrix Algebra

I am trying to solve the following question. Let $A$ be the algebra over $\mathbb{C}$ consisting of matrices of the form \begin{pmatrix} * & * & 0 & 0 \\ * & * & 0 & 0 \\ * & * & * & 0 \\ * & * & * & * \\ \end{pmatrix} […]

Is every regular element of a ring invertible?

When I am reading a paper, I found the definition of a new ring as following: In this definition, if every central regular element is invertible, i.e., how to understand the invertible element of u? how to prove it is a ring?Moreover, is the product of a regular element and a unit also a unit? […]

Nilpotent elements of a non-commutative ring with trivial automorphism group form an ideal

Let $R$ be a non-commutative ring with identity such that the identity map is the only ring automorphism of $R$. Prove that the set $N$ of all nilpotent elements of $R$ is an ideal of $R$.

Ideal in a matrix ring

Edit As Kimball point out, in the following question for me an ideal $I$ is a full $\mathbb{Z_p}$-lattice of $M_2(\mathbb{Q}_p)$ such that $$\lbrace\alpha \in M_2(\mathbb{Q}_p)\vert \alpha I \subset I \rbrace=M_2(\mathbb{Z}_P). $$ Let $p$ be a prime number and consider the ring, formed by the elements $$ \begin{pmatrix} a & b \\ pc & d \end{pmatrix}$$ […]

Do Lipschitz/Hurwitz quaternions satisfy the Ore condition?

The Lipschitz quaternions $L$ are the quaternions with integer coefficients and the Hurwitz quaternions $H$ are the quaternions with coefficients from $\Bbb Z\cup(\Bbb Z+1/2).$ A ring satisfies the right Ore condition when for $x,y\in R$ the right principal ideals generated by them have a non-empty intersection. I can infer from what I’ve seen in various […]

“Vector spaces” over a skew-field are free?

Are modules over a skew field free? That is, if $F$ is a skewfield then can any module $M$ be written as $\underset{i \in I}{\bigoplus} F$ for some indexing set $I$?

Prime which is not irreducible in non-commutative ring with unity without zero divisors

In a non-commutative ring with unity without zero divisors find a prime element which is not irreducible (if possible). $p$ is prime iff $p|ab$ implies that $p|a$ or $p|b$, and $x$ is irreducible iff $x = ab$ implies that either $a$ or $b$ is a unit.

Lifting idempotents modulo a nilpotent ideal

The problem is this: Suppose $I \subseteq R$ is a nilpotent ideal and there is $r \in R$ with $r \equiv r^2 \pmod I$. Show $r \equiv e \pmod I$ for some $e \in R$ idempotent. I have spent a few hours rolling around in abstracta with no destination. I believe that if I could […]