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...

For every integer d ≥ 1, there is a unital closed subalgebra A d ⊂ B(H) with similarity degree equal precisely to d, in the sense of our previous paper. This means that for any unital homomorphism u: A d → B(H) we have u cb ≤ Ku d with K > 0 independent of u, and the exponent d in this estimate cannot be improved. The proof that the degree is larger than d − 1 crucially uses an upper bound for ...

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

This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S = {x ∈ R : g1(x) ≥ 0, · · · , gs(x) ≥ 0} modulo its KKT (Karush-KuhnTucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x) > 0 on S; further...

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...

Finding the key of symmetric cipher takes computing common zero of polynomials, which de ne ideal and corresponding variety, usually considered over algebraically closed eld. The solution is the point of the variety over prime eld; it is de ned by a sum of the polynomial ideal and the eld ideal that de nes prime eld. Some authors use partitioning of this sum and reducing syzygies of polynomial ...