### $*$-biregular rings

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Let $R$ be a ring and $M$ a right $R$-module. $M$ is called $\oplus $-cofinitely supplemented if every submodule $N$ of $M$ with $\frac{M}{N}$ finitely generated has a supplement that is a direct summand of $M$. In this paper various properties of the $\oplus $-cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of $\oplus $-cofinitely supplemented modules is $\oplus $-cofinitely supplemented. (2) A ring $R$ is semiperfect if and only if every free $R$-module is $\oplus $-cofinitely supplemented. In addition, if $M$ has the summand sum...

The duals of $\cup $-compact modules are briefly discussed.

We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of ${\aleph}_{k}$-free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is ${\aleph}_{k}$-free if every subset of size $<{\aleph}_{k}$ is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a module G is...

It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $\left(1\right)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $(n,d)$-injective (pre)cover; $\left(2\right)$ if $R$ is a ring such that every $(n,0)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $(n,0)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $(n,d)$-rings, von Neumann regular rings and semisimple rings....

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example [1], [2], [3], [4], [5] and [8]) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like behaviour. In this paper we generalize some of the results due to Bell and Mason [4] on near-rings admitting a special type of derivation...

Let $\mathcal{R}$ be a semiprime ring with unity $e$ and $\phi $, $\varphi $ be automorphisms of $\mathcal{R}$. In this paper it is shown that if $\mathcal{R}$ satisfies $$2\mathcal{D}\left({x}^{n}\right)=\mathcal{D}\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)\mathcal{D}\left(x\right)+\mathcal{D}\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)\mathcal{D}\left({x}^{n-1}\right)$$ for all $x\in \mathcal{R}$ and some fixed integer $n\ge 2$, then $\mathcal{D}$ is an ($\phi $, $\varphi $)-derivation. Moreover, this result makes it possible to prove that if $\mathcal{R}$ admits an additive mappings $\mathcal{D},\mathcal{G}:\mathcal{R}\to \mathcal{R}$ satisfying the relations $$\begin{array}{c}2\mathcal{D}\left({x}^{n}\right)=\mathcal{D}\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)\mathcal{G}\left(x\right)+\mathcal{G}\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)\mathcal{G}\left({x}^{n-1}\right)\phantom{\rule{0.166667em}{0ex}},\\ 2\mathcal{G}\left({x}^{n}\right)=\mathcal{G}\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)\mathcal{D}\left(x\right)+\mathcal{D}\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)\mathcal{D}\left({x}^{n-1}\right)\phantom{\rule{0.166667em}{0ex}},\end{array}$$ for all $x\in \mathcal{R}$ and some fixed integer $n\ge 2$, then $\mathcal{D}$ and $\mathcal{G}$ are ($\phi $, $\varphi $)derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.

The class of almost completely decomposable groups with a critical typeset of type (1,4) and a homocyclic regulator quotient of exponent p³ is shown to be of bounded representation type. There are precisely four near-isomorphism classes of indecomposables, all of rank 6.

The main result of the note is a characterization of 1-amenability of Banach algebras of approximable operators for a class of Banach spaces with 1-unconditional bases in terms of a new basis property. It is also shown that amenability and symmetric amenability are equivalent concepts for Banach algebras of approximable operators, and that a type of Banach space that was long suspected to lack property 𝔸 has in fact the property. Some further ideas on the problem of whether or not amenability (in...

We investigate 2-local Jordan automorphisms on operator algebras. In particular, we show that every 2-local Jordan automorphism of the algebra of all n× n real or complex matrices is either an automorphism or an anti-automorphism. The same is true for 2-local Jordan automorphisms of any subalgebra of ℬ which contains the ideal of all compact operators on X, where X is a real or complex separable Banach spaces and ℬ is the algebra of all bounded linear operators on X.

Let X and Y be complex Banach spaces of dimension greater than 2. We show that every 2-local Lie isomorphism ϕ of B(X) onto B(Y) has the form ϕ = φ + τ, where φ is an isomorphism or the negative of an anti-isomorphism of B(X) onto B(Y), and τ is a homogeneous map from B(X) into ℂI vanishing on all finite sums of commutators.

2000 Mathematics Subject Classification: 16R10, 16R20, 16R50The algebra Mn(K) of the matrices n × n over a field K can be regarded as a Z-graded algebra. In this paper, it is proved that if K is an infinite field, all the Z-graded polynomial identities of Mn(K) follow from the identities: x = 0, |α(x)| ≥ n, xy = yx, α(x) = α(y) = 0, xyz = zyx, α(x) = −α(y) = α(z ), where α is the degree of the corresponding variable. This is a generalization of a result of Vasilovsky about the Z-graded identities...