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In this note we prove that the -points of the representation Artin-stack of n-dimensional representations of an affine ℂ-algebra R correspond to ℂ-algebra morphisms R → A n where A n is an Azumaya algebra of degree n over C. We connect this to the theory of D-branes and Azumaya noncommutative geometry, developed by Chien-Hao Liu and Shing-Tung Yau in a series of articles [9]–[14]. [ABSTRACT FROM PUBLISHER]

Some recent results of Ayache on going-down domains and extensions of domains that either are residually algebraic or have DCC on intermediate rings are generalized to the context of extensions of commutative rings. Given a finite maximal chain 𝒞 of R-subalgebras of a weak Baer ring T, it is shown how a “min morphism” hypothesis can be used to transfer the “going-down ring” property from R to each member of 𝒞. The integral minimal ring extensions which are min morphisms are classified. The ring extensions satisfying FCP (i.e., for which each chain of intermediate rings is finite) are characterized as the strongly affine extensions with DCC on intermediate rings. In the relatively integrally closed case, such extensions R ⊆ T induce open immersions Spec(S) → Spec(R) for each R-subalgebra S of T. [ABSTRACT FROM PUBLISHER]

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands. [ABSTRACT FROM PUBLISHER]

Here we study the structure of a group with the presentation by conjugation which has been assigned to an extended affine root system R. We show that is isomorphic to the direct product of the extended affine Weyl group 𝒲 of R and the kernel of a natural epimorphism . The article also shows that has structural features that are well known for 𝒲. [ABSTRACT FROM PUBLISHER]

This article gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. This point of view makes many relationships between the virtual braid group and the pure virtual braid group apparent, and makes representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter Equation equally transparent. In this categorical framework, the virtual braid group has nothing to do with the plane and nothing to do with virtual crossings. It is a natural group associated with the structure of algebraic braiding. [ABSTRACT FROM PUBLISHER]

We investigate the least studied class of differential rings—the class of differential rings of nonzero characteristic. We present the notion of differentially closed quasifield and develop geometrical theory of differential equations in nonzero characteristic. The notions of quasivariety and its morphisms are scrutinized. Presented machinery is a basis for reduction modulo p for differential equations. [ABSTRACT FROM PUBLISHER]

We study deformations of Leibniz algebra morphisms over a commutative local algebra base with 1. We construct the associated deformation cohomology that controls deformations using the cochain complex defining the Leibniz cohomology. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field, B be a finite dimensional k-algebra and A be the one-point extension of B by the projective B-module P0. We compare the posets TA and TB of tilting A-modules and B-modules, respectively. We prove that the restriction and the extension functors define morphisms of posets r: TA → TB and e: TB → TA such that re = id. Moreover, e induces a full embedding of the quiver of TB into that of TA, whose image is closed under successors, and mapping distinct connected components of the first into distinct connected components of the second. [ABSTRACT FROM AUTHOR]

In this note,we answer a question of Hong et al. (2003) by proving that if a is a monomorphism of a reduced ring R , and R is a-skew Armendariz, then R is a-rigid. [ABSTRACT FROM AUTHOR]