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We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations and over the quaternion algebra ℍ, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations over ℍ to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied. [ABSTRACT FROM AUTHOR]

In this paper, the derivation algebra and the automorphism group of the twisted N=2 superconformal algebra are completely determined. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new example of Hopf group coalgebras by considering Radford's biproduct Hopf algebra in the Turaev category. Furthermore, we find some sufficient and necessary conditions for such Radford's biproduct Hopf algebra to admit quasitriangular structures in the sense of Turaev group coalgebras. [ABSTRACT FROM AUTHOR]

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ∗-clean, where R is a commutative local ring and G is one of the groups C3, C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ∗-clean. [ABSTRACT FROM AUTHOR]

In the present paper we describe a class of Φ-Auslander-Yoneda algebras over K[x]/(xn) in terms of quivers with relations, and prove that they are actually cellular algebras in the sense of Graham and Lehrer. [ABSTRACT FROM AUTHOR]

In this paper, we obtain (non-commutative) Gröbner-Shirshov bases for rectangular bands, small extensions of semigroups and Bruck-Reilly extensions of monoids in terms of the general product. [ABSTRACT FROM AUTHOR]