Functional Conceptual Substratum as a New Cognitive Mechanism for Mathematical Creation

We describe a new cognitive ability, i.e., functional conceptual substratum, used implicitly in the generation of several mathematical proofs and definitions. Furthermore, we present an initial (first-order) formalization of this mechanism together with its relation to classic notions like primitive positive definability and Diophantiveness. Additionally, we analyze the semantic variability of functional conceptual substratum when small syntactic modifications are done. Finally, we describe mathematically natural inference rules for definitions inspired by functional conceptual substratum and show that they are sound and complete w.r.t. standard calculi

Normality and Related Properties of Forcing Algebras

We present a sufficient condition for irreducibility of forcing algebras and study the (non)-reducedness phenomenon. Furthermore, we prove a criterion for normality for forcing algebras over a polynomial base ring with coefficients in a perfect field. This gives a geometrical normality criterion for algebraic (forcing) varieties over algebraically closed fields. Besides, we examine in detail an specific (enlightening) example with several forcing equations. Finally, we compute explicitly the normalization of a particular forcing algebra by means of finding explicitly the generators of the ideal defining it as an affine ring

Towards an Homological Generalization of the Direct Summand Theorem

We present a more general (parametric-) homological characterization of the Direct Summand Theorem. Specifically, we state two new conjectures: the Socle-Parameter conjecture (SPC) in its weak and strong forms. We give a proof for the week form by showing that it is equivalent to the Direct Summand Conjecture (DSC), now known to be true after the work of Y. Andr\'{e}, based on Scholze's theory of perfectoids. Furthermore, we prove the SPC in its strong form for the case when the multiplicity of the parameters is smaller or equal than two. Finally, we present a new proof of the DSC in the equicharacteristic case, based on the techniques thus developed

On the local metric dimension of corona product graphs

A vertex $v\in V(G)$ is said to distinguish two vertices $x,y\in V(G)$ of a nontrivial connected graph $G$ if the distance from $v$ to $x$ is different from the distance from $v$ to $y$. A set $S\subset V(G)$ is a local metric generator for $G$ if every two adjacent vertices of $G$ are distinguished by some vertex in $S$. A local metric generator with the minimum cardinality is called a local metric basis for $G$ and its cardinality, the local metric dimension of G. In this paper we study the problem of finding exact values for the local metric dimension of corona product of graphs

Generalizations of the Direct Summand Theorem over UFD-s for some Bigenerated Extensions and an Asymptotic Version of Koh's Conjecture

This article deals with two different problems in commutative algebra. In the first part, we give a proof of generalized forms of the Direct Summand Theorem (DST (or DCS)) for module-finite extension rings of mixed characteristic $R\subset S$ satisfying the following hypotheses: The base ring $R$ is a Unique Factorization Domain of mixed characteristic zero. We assume that $S$ is generated by two elements which satisfy, either radical quadratic equations, or general quadratic equations under certain arithmetical restrictions. In the second part of this article, we discuss an asymptotic version of Koh's Conjecture. We give a model theoretical proof using "non-standard methods".Comment: Updated version after the result of Y. Andre regarding the DS

A General Version of the Nullstellensatz for Arbitrary Fields

We prove a general version of Bezout's form of the Nullstellensatz for arbitrary fields. The corresponding sufficient and necessary condition only involves the local existence of multi-valued roots for each of the polynomials belonging to the ideal in consideration. Finally, this version implies the standard Nullstellensatz when the coefficient field is algebraically closed

Containment-Division Rings and New Characterizations of Dedekind Domains

We introduce a new class of commutative rings with unity, namely, the Containment-Division Rings (CDR-s). We show that this notion has a very exceptional origin since it was essentially co-discovered with the qualitative help of a computer program (i.e. The Heterogeneous Tool Set (HETS)). Besides, we show that in a Noetherian setting, the CDR-s are just another way of describing Dedekind domains. Simultaneously, we see that for CDR-s, the Noetherian condition can be replaced by a weaker Divisor Chain Condition

On Preservation Properties and a Special Algebraic Characterization of Some Stronger Forms of the Noetherian Condition

We give an elementary proof prove of the preservation of the Noetherian condition for commutative rings with unity $R$ having at least one finitely generated ideal $I$ such that the quotient ring is again finitely generated, and $R$ is $I-$adically complete. Moreover, we offer as a direct corollary a new elementary proof of the fact that if a ring is Noetherian then the corresponding ring of formal power series in finitely many variables is Noetherian. In addition, we give a counterexample showing that the `completion' condition cannot be avoided on the former theorem. Lastly, we give an elementary characterization of Noetherian commutative rings that can be decomposed as a finite direct product of fields

Second cohomology group of the finite-dimensional simple Jordan superalgebra Dt\mathcal{D}_{t}, t≠0t\neq 0

The second cohomology group (SCG) of the Jordan superalgebra $\mathcal{D}_{t}$, $t\neq 0$, is calculated by using the coefficients which appear in the regular superbimodule $\mathrm{Reg}\mathcal{D}_t$. Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem \cite{Faber1}. First, to calculate the SCG of a Jordan superalgebra we use split-null extension of the Jordan superalgebra and the Jordan superalgebra representation. We prove conditions that satisfy the bilinear forms $h$ that determine the SCG in Jordan superalgebras. We use these to calculate the SCG for the Jordan superalgebra $\mathcal{D}_{t}$ , $t\neq 0$. Finally, we prove that $\mathcal{H}^2(\mathcal{D}_{t}, \textrm{Reg}\mathcal{D}_{t})=0\oplus\mathbb{F}^2$, $t\neq 0$.Comment: 10 page

Jordan super algebras of type JPnJP_n, n≥3n\geq 3 and the Wedderburn principal theorem

We investigate an analogue to the Wedderburn Principal Theorem (WPT) for a finite-dimensional Jordan superalgebra $J$ with solvable radical $N$ such that $N^2=0$ and $J/N\cong JP_n$, $n\geq 3$. We consider $N$ as an irreducible $JP_n$-bimodule and we prove that the WPT holds for $J$.Comment: 12 page