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A first-order theory is Noetherian with respect to the collection of formulae $\mathcal{F}$ if every definable set is a Boolean combination of instances of formulae in $\mathcal{F}$ and the topology whose subbasis of closed sets is the…
We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categories arises in this way. Similar factorisation systems…
We obtain a complete classification of minimal simple unitary $W$-algebras.
A ring $R$ is called left strictly $(<\aleph_{\alpha})$-noetherian if $\aleph_{\alpha}$ is the minimum cardinal such that every ideal of $R$ is $(<\aleph_{\alpha})$-generated. In this note, we show that for every singular (resp., regular)…
Let A and B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra that contains A. Denote by A' the integral closure of A in B. We show that A' is determined by finitely many unique discrete valuation rings.…
The algebraic theory of third-order tensors under the $t$-product is naturally formulated over the complex field via Fourier block diagonalization. However, many applications require real-valued representations. In this paper, we…
We study the Frattini subalgebra of Leibniz algebras generated by one element. We also investigate Leibniz algebras all of whose proper subalgebras are elementary.
The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as…
Recently, Ramos and Whiting showed that any generalized cluster algebra of geometric type is isomorphic to a quotient of a subalgebra of a certain cluster algebra. Based on their idea and method, we show that the same property holds for any…
We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In…
We classify all division algebras that are principal Albert isotopes of a cyclic Galois field extension of degree $n>2$ up to isomorphisms. We achieve a ``tight'' classification when the cyclic Galois field extension is cubic. The…
Let $R$ be a semilocal geometrically factorial Noetherian domain of characteristic zero. We show that a reductive $R$-group scheme is isotropic if it is generically isotropic. We derive various consequences, in particular for the…
An algebraizable singularity is a germ of a singular holomorphic foliation which can be defined in some appropriate local chart by a differential equation with algebraic coefficients. We show that there exists at least countably many…
We classify the factorizations of finite classical groups with nonsolvable factors, completing the classification of factorizations of finite almost simple groups.
We give a precise description of how the class group of a number field measures the failure of unique factorization in its ring of integers. Specifically, following ideas of Kummer, we determine the structure of all irreducible…
There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there…
We define Khovanov-Lauda-Rouquier subalgebras which are generalizations of redotted versions of Webster's tensor product algebras of type $A_1$. Quotient algebras of these subalgebras are isomorphic to Webster's tensor product algebras in…
We functorially characterize groupoids as special dagger Frobenius algebras in the category of sets and relations. This is then generalized to a non-unital setting, by establishing an adjunction between H*-algebras in the category of sets…
We describe the T-space of central polynomials for both the unitary and the nonunitary finite dimensional Grassmann algebra over a field of characteristic p not equal to 2 (infinite field in the case of the unitary algebra).
We find necessary and sufficient conditions for a complete local ring containing the rationals to be the completion of a countable excellent local (Noetherian) domain. Furthermore, we find necessary and sufficient conditions for a complete…