We show that any non-type I separable unital AF algebra B can be modeled from inside by a nonnuclear C*-algebra and from outside by a nonexact C*-algebra. More precisely there exist unital separable quasidiagonal C*-algebras A B C of real rank zero, stable rank one, such that A is nonnuclear, C is nonexact, and both A and C are asymptotically homotopy equivalent to B. In particular A, B and C h...

In this article we obtain 2 generalizations of the well known Gleason-Kahane-Z̀elazko Theorem. We consider a unital Banach algebra A, and a continuous unital linear mapping φ of A into Mn(C) – the n × n matrices over C. The first generalization states that if φ sends invertible elements to invertible elements, then the kernel of φ is contained in a proper two sided closed ideal of finite codimen...

We obtain several fundamental results on finite index ideals and additive subgroups of rings as well model-theoretic connected components rings, which concern generating in finitely many steps inside groups rings. Let R be any ring equipped with an arbitrary additional first order structure, A a set parameters. show that whenever H is A-definable, subgroup (R,+), then H+R⋅H contains two-sided i...

We call a local vector lattice any vector lattice with a distinguished positive strong unit and having exactly one maximal ideal (its radical). We provide a short study of local vector lattices. In this regards, some characterizations of local vector lattices are given. For instance, we prove that a vector lattice with a distinguished strong unit is local if and only if it is clean with non no-...

It has been established by Inoue that a complex locally C*-algebra with a dense ideal posesses a bounded approximate identity which belonges to that ideal. It has been shown by Fritzsche that if a unital complex locally C*-algebra has an unbounded element then it also has a dense one-sided ideal. In the present paper we obtain analogues of the aforementioned results of Inoue and Fritzsche for r...

We consider extensions of unital commutative rings. We define an extension R ↪→ S to be a p-extension if every principally generated ideal of S is generated by an element of R. Examples are plentiful and localizations of regular multiplicative sets are p-extensions. We develop the theory of pextensions.

Let R be a commutative ring with identity and M be a unital R-module. Then M is called a multiplication module provided for every submodule N of M there exists an ideal I of R such that N = IM. Our objective is to investigate properties of prime and semiprime submodules of multiplication modules. Mathematics Subject Classification: 13C05, 13C13

We consider extensions of unital commutative rings. We define an extension R ↪→ S to be a p-extension if every principally generated ideal of S is generated by an element of R. Examples are plentiful and localizations of regular multiplicative sets are p-extensions. We develop the theory of pextensions.

Using different descriptions of the Cuntz semigroup and Pedersen ideal, it is shown that $\sigma$-unital simple $C^*$-algebras with almost unperforated semigroup, a unique lower semicontinuous $2$-quasitrace whose stabilization has stable rank $1$ are either or algebraically simple.