Articles of noncommutative algebra

In the lattice of ideals, what are the lowerbounds of the prime ideals?

Take the lattice of ideals in a non-commutative ring with 1. It is well known that the lowerbounds of all the maximal ideals are the superfluous ideals. Is there a similar characterization for the lowerbounds of all the prime ideals? They must be close to the sum of nilpotent ideals but I can’t seem to […]

A prime ring whose socle is nonzero and of finite length is simple Artinian.

This is a part of an exercise (Sect. 14 Exercise 11) in Anderson & Fuller’s book “Rings and Categories of Modules”, and I’m a graduate level student in Turkey. I want to prove that if $R$ is a prime ring with $0\not=Soc(_RR)$ which is of finite length, then $R$ is a simple Artinian ring. I […]

A doubt about lower nil radical while proving 2-primality of ring.( Baer-McCoy Radical)

I was proving that a reversible ring is 2-Primal for an exercise in T.Y Lam’s book, but I got stuck. Here is where I’m stuck: let $a$ be a nilpotent element of $R$ with $a^n=0$. Then using reversibility of ring he concluded $(RaR)^n=0$, but then he says this implies that $RaR\subset \mathrm{Nil}_\ast(R)$ which clearly gives […]

Is the tensor product of non-commutative algebras a colimit?

For $R$ a commutative ring, the tensor product of $R$-algebras is the coproduct in the category of commutative $R$-algebras. In the noncommutative case it is no longer the coproduct in the category of associative $R$-algebras, but it does satisfy a universal property, as given on Wikipedia. Is this some sort of colimit? If not, is […]

Show that $I$ is an ideal

Let $R$ be a ring and $I\subseteq R$ the only maximal right ideal of $R$. I want to show that $I$ is an ideal. To show that $I$ is an ideal, we have to show that $I$ is a left ideal, right? How could we show that? Could you give me some hints?

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 $R$.