### A cluster expansion formula (${A}_{n}$ case).

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If $A{\otimes}_{R,\sigma}V$ and $A{\otimes}_{P,\nu}W$ are two Brzeziński crossed products and Q: W⊗ V → V⊗ W is a linear map satisfying certain properties, we construct a Brzeziński crossed product $A{\otimes}_{S,\theta}\left(V\otimes W\right)$. This construction contains as a particular case the iterated twisted tensor product of algebras.

For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$; the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$. Its maximal cones are naturally indexed by the $c$-sortable elements of $W$. The main result of this paper is that the known bijection cl${}_{c}$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for $W$. The rays...

Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite-dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection...

The main goal of this paper is to give a mathematical foundation, serious and consistent, to some parts of Santillis isotheory. We study the isotopic liftings of rings, subrings and ideals, and we also introduce the concept of quotient isoring. By using the isoproduct model, necessary conditions assuring the existence of such isostructures are given. Such conditions are based on the inner laws which originate the associated elements of isotopy. These elements will allow to extend, from a different...

The main result of this paper is the introduction of a notion of a generalized $R$-Latin square, which includes as a special case the standard Latin square, as well as the magic square, and also the double stochastic matrix. Further, the algebra of all generalized Latin squares over a commutative ring with identity is investigated. Moreover, some remarkable examples are added.

An $\mathcal{S}$-closed submodule of a module $M$ is a submodule $N$ for which $M/N$ is nonsingular. A module $M$ is called a generalized CS-module (or briefly, GCS-module) if any $\mathcal{S}$-closed submodule $N$ of $M$ is a direct summand of $M$. Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right $R$-modules are projective if and only if all right $R$-modules are GCS-modules.

A class of semirings, so called p-semirings, characterized by a natural number p is introduced and basic properties are investigated. It is proved that every p-semiring is a union of skew rings. It is proved that for some p-semirings with non-commutative operations, this union contains rings which are commutative and possess an identity.

In this note, for a ring endomorphism $\alpha $ and an $\alpha $-derivation $\delta $ of a ring $R$, the notion of weakened $(\alpha ,\delta )$-skew Armendariz rings is introduced as a generalization of $\alpha $-rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(\alpha ,\delta )$-skew Armendariz ring if and only if ${T}_{n}\left(R\right)$ is weakened $(\overline{\alpha},\overline{\delta})$-skew Armendariz if and only if $R\left[x\right]/\left({x}^{n}\right)$ is weakened $(\overline{\alpha},\overline{\delta})$-skew Armendariz ring for any positive integer $n$.