I-Nearly Prime Submodules

Let be a commutative ring with identity, and a fixed ideal of and be an unitary -module. In this paper we introduce and study the concept of -nearly prime submodules as genrealizations of nearly prime and we investigate some properties of this class of submodules. Also, some characterizations of -nearly prime submodules will be given.

I-Semiprime Submodules

Let be a commutative ring with identity and a fixed ideal of and be an unitary -module.We say that a proper submodule of is -semi prime submodule if with . In this paper, we investigate some properties of this class of submodules. Also, some characterizations of -semiprime submodules will be given, and we show that under some assumptions -semiprime submodules and semiprime submodules are coincided.

MODEL OF SOCIAL POTENTIAL OF CHILDHOOD IN THE RUSSIAN REGIONS: THE RESULTS OF EXPERIMENTS

The article presents the results of mathematical experiments with the system «Social potential of childhood in the Russian regions». In the structure of system divided into three subsystems – the «Reproduction of children in the region», «Children’s health» and «Education of children», for each defined its target factor (output parameter). The groups of infrastructure factors (education, health, culture and sport, transport), socio-economic, territorial-settlement, demographic and en-vironmental factors are designated as the factors that control the system (input parameters). The aim of the study is to build a model îf «Social potential of childhood in the Russian regions», as well as to conduct experiments to find the optimal ratio of the values of target and control factors. Three waves of experiments were conducted. The first wave is related to the analysis of the dynam-ics of indicators for 6 years. The second – with the selection of optimal values of control factors at fixed ideal values of target factors. The third wave allowed us to calculate the values of the target factors based on the selected optimal values of the control factors of the previous wave.

A construction of Prüfer rings involving quotients of Rees algebras

A family of quotients of a Rees algebra associated to a ring with respect to a fixed ideal was recently introduced by Barucci, D’Anna and Strazzanti. In this paper, we will classify rings of this family that satisfy certain Prüfer-like properties and, as a particular case, we will extend results obtained for amalgamated duplications and Nagata idealizations.

Making and Breaking Mathematical Sense

In line with the emerging field of philosophy of mathematical practice, this book pushes the philosophy of mathematics away from questions about the reality and truth of mathematical entities and statements and toward a focus on what mathematicians actually do—and how that evolves and changes over time. How do new mathematical entities come to be? What internal, natural, cognitive, and social constraints shape mathematical cultures? How do mathematical signs form and reform their meanings? How can we model the cognitive processes at play in mathematical evolution? And how does mathematics tie together ideas, reality, and applications? This book combines philosophical, historical, and cognitive studies to paint a fully rounded image of mathematics not as an absolute ideal but as a human endeavor that takes shape in specific social and institutional contexts. The book builds on case studies to confront philosophical reconstructions and cutting-edge cognitive theories. It focuses on the contingent semiotic and interpretive dimensions of mathematical practice, rather than on mathematics' claim to universal or fundamental truths, in order to explore not only what mathematics is, but also what it could be. Along the way, the book challenges conventional views that mathematical signs represent fixed, ideal entities; that mathematical cognition is a rigid transfer of inferences between formal domains; and that mathematics' exceptional consensus is due to the subject's underlying reality. The result is a revisionist account of mathematical philosophy that will interest mathematicians, philosophers, and historians of science alike.

Principal Mappings between Posets

We introduce and study principal mappings between posets which generalize the notion of principal elements in a multiplicative lattice, in particular, the principal ideals of a commutative ring. We also consider some weaker forms of principal mappings such as meet principal, join principal, weak meet principal, and weak join principal mappings which also generalize the corresponding notions on elements in a multiplicative lattice, considered by Dilworth, Anderson and Johnson. The principal mappings between the lattices of powersets and chains are characterized. Finally, for any PIDR, it is proved that a mappingF:Idl(R)→Idl(R)is a contractive principal mapping if and only if there is a fixed idealI∈Idl(R)such thatF(J)=IJfor allJ∈Idl(R). This exploration also leads to some new problems on lattices and commutative rings.

Alternative polarizations of Borel fixed ideals

For a monomial idealIof a polynomial ringS, apolarizationofIis a square-free monomial idealJof a larger polynomial ringsuch thatS/Iis a quotient of/Jby a (linear) regular sequence. We show that a Borel fixed ideal admits a nonstandard polarization. For example, while the usual polarization sendsours sends it tox1y2y3Using this idea, we recover/refine the results onsquare-free operationin the shifting theory of simplicial complexes. The present paper generalizes a result of Nagel and Reiner, although our approach is very different.

Elimination from Homogeneous Polynomials Over a Polynomial Ring

Let Ω be a field and Γ a parameter. We designate the set of all polynomials homogeneous in (X) = (X1, … , Xn) with coefficients in Ω [Γ] by H Ω Γ[X] and write such polynomials as F, F(X), or F(X, Γ). The degree of a polynomial in H Ω Γ [X] shall mean the degree in (X). Let I = (F1 … , Fr) be a fixed ideal in H Ω Γ [X] generated by F1 … , Fr.