Two applications of Nagata rings and modules

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].

THE PROBLEMS OF THEORETIC NEUROLOGY: INFORMATION-COMMUTATIVE THEORY OF HUMAN BRAIN AND PRINCIPLES OF ITS FUNCTIONING

The article summarizes contemporary scientific concepts of brain organization and structure. For the first time we developed information principles of brain functioning and the basics of information communicative theory of brain. Here we demonstrate that information commutative organization (ICO) of brain has multilevel structure of open type represented by the block of information capture and transfer (BICT) and the block of information processing (BIP). BICT is a switchboard with information dispatcher and router in nervous tissue (NT). BICT has system vertical and complex horizontal commutation. Brain ICPs are presented as interthecal information registers (ITIRs) that provide extraneural information processing in subarachnoid space (SAS) and subdural space (SDS) above brain. The arcs of conditioned and unconditioned reflexes close in SAS extraneurally when electro-magnetic waves (EMWs) of cortical ICM subscribers are reflected from the arachnoid mater and information is automatically sent to ICM receivers of brain cortex (reflectory ITIRs). Information processing (analysis, synthesis, calculations) is based on dissipation of EMWs of cortical NT ICMs in SAS and SDS cerebrospinal fluid (CSF) forming support and object EMWs, which, being reflected from the arachnoid or dura mater form holograms of information images matrixes (IIMs) by interference, diffraction and superimposition while IIMs are the basics of brain cognitive functions (cognitive ITIRs). The theory of brain can become a new milestone in the development of therapies of nervous diseases, neuromorphic computation, innovative systems of artificial intellect and novel approaches to brain-computer interface.

Conditional expectation and stochastic integrals in non-commutative Lp spaces

The aim of the paper is to propose a general scheme for the consideration of non-commutative stochastic integrals. The role of a probability space is played by a couple (, φ0), where is a von Neumann algebra and φ0 is a faithful normal state on . Our processes live in the algebra of all measurable operators associated with the crossed product of by the modular automorphism group The algebra contains all the (Haagerup's) Lp spaces over . The measure topology of the algebra has the nice feature of inducing the Lp norm topology on the Lp spaces, which makes it particularly suitable for defining stochastic integrals. The commutative theory fits smoothly into the scheme, although there exists no canonical way of embedding the algebra of (commutative) random variables into . In fact, for any commutative stochastic process we have a family of different non-commutative stochastic processes corresponding to the process. This arbitrariness seems to be quite natural in the non-commutative context. An appropriate example can be found at the end of the paper (Section 6, C4).

On Central Ω-Krull Rings and their Class Groups

The aim of this note is to study the class group of a central Ω-Krull ring and to determine in some cases whether a twisted (semi) group ring is a central Ω-Krull ring. In [8] we defined an Ω-Krull ring as a generalization of a commutative Krull domain. In the commutative theory, the class group plays an important role. In the second and third section, we generalize some results to the noncommutative case, in particular the relation between the class group of a central Ω-Krull ring and the class group of a localization. Some results are obtained in case the ring is graded. Theorem 3.2 establishes the relation between the class group and the graded class group. In particular, in the P.I. case we obtain that the class group is equal to the graded class group. As a consequence of a result on direct limits of Ω-Krull rings, we are able to derive a necessary and sufficient condition in order that a polynomial ring R[(Xi)i∊I] (I may be infinite) is a central Ω-Krull ring.

A class of hyperrings and hyperfields

Hyperring is a structure generalizing that of a ring, but where the addition is not a composition, but a hypercomposition, i.e., the sumx+yof two elements,x,y, of a hyperringHis, in general, not an element but a subset ofH. When the non-zero elements of a hyperring form a multiplicative group, the hyperring is called a hyperfield, and this structure generalizes that of a field. A certain class of hyperfields (residual hyperfields of valued fields) has been used by the author [1] as an important technical tool in his theory of approximation of complete valued fields by sequences of such fields. Tne non-commutative theory of hyperrings (particularly Artinian) has been studied in depth by Stratigopoulos [2].The question arises: How common are hyperrings? We prove in this paper that a conveniently defined quotientR/Gof any ringRby any normal subgroupGof its multiplicative semigroup is always a hyperring which is a hyperfield whenRis a field. We ask: Are all hyperrings isomorphic to some subhyperring of a hyperring belonging to the class just described?