广义α—结合BCI—代数

引入了广义α－结合ＢＣＩ－代数的概念，研究了ＢＣＩ－代数的ｐ－半单部分与广义α－结合部分的关系。并将ｐ－半单ＢＣＩ－代数的若干重要性质推广到广义α－结合ＢＣＩ－代数上。最后我们证明了每个广义α－结合ＢＣＩ－代数可确定一个交换偏序幺半群。本文结果表明文［１］的正则ＢＣＩ－代数与ｐ－半单ＢＣＩ－代数是一致的。     

关于单BCI－代数的一些结果

本文讨论了单ＢＣＩ－代数．证明了一个ＢＣＩ－代数是单的当且仅当它的子代数都是单的；给出了单ｐ－半单ＢＣＩ－代数的一种表示式；证明了一个ｐ－半单ＢＣＩ－代数是单的当且仅当它的阶是素数；这样得到了一批（无限多个）单ＢＣＩ－代数；证明了商ＢＣＫ－代数Ｘ／Ａ是单的当且仅当Ａ是Ｘ的极大理想．     

关于BCI—代数伴随半群中的可逆元

给出了ＢＣＩ－代数的伴随半群中可逆元的刻划，说明ＢＣＩ－代数的ｐ－半单闭理想与其伴随半群的子群之间有一一对应关系，并证明最大的ｐ－半单闭理想是根理想。     

半单优BCI－代数的长度定理

继续对［１］的研究，得到了半单优ＢＣＩ－代数的替换定理和长度定理．     

正定关联BCI—代数

本文是作者［１］和［２］的继续，引入了正定关联ＢＣＩ－代数的概念，并证明了：正定关联ＧＢＣＫ－代数类和Ｐ－半单ＢＣＩ－代数类是正定关联ＢＣＩ－代数类的真子类。     

半单优BCI—代数的长度定理

继续对「１」的研究，得到了半单优ＢＣＩ－代数的替换定理和长度定理。     

BCI—代数的KI并代数

本文引入ＢＣＩ－代数的ＫＩ并代数，它是ＢＣＫ－代数的无交并的推广。     

BCI—代数和弱半联理想

引进ＢＣＩ－代数的弱关联理想的概念，并用以刻划弱关联ＢＣＩ－代数，从而推广了文［３］中的的一上结结论。     

BCI—代数的Fuzzy强蕴涵理想

本文引入了ＢＣＩ－代数的强蕴函理想的概念，研究了它的性质，同时讨论了ＢＣＩ－代数中的Ｆｕｚｚｙ关系。     

路代数的有向图特征与Brown－McCoy根

本文解决路代数中若干遗留问题，给出本原路代数、（右）Ｇｏｌｄｉｅ路代数的有向图特征，证明广义路代数的Ｂｒｏｗｎ－ＭｃＣｏｙ根与它的Ｊａｃｏｂｓｏｎ根不必重合．     

Modules and ideals of algebras of associative type

In this paper, we study some properties of algebras of associative type introduced in previous papers of the author. We show that a finite-dimensional algebra of associative type over a field of zero characteristic is homogeneously semisimple if and only if a certain form defined by the trace form is nonsingular. For a subclass of algebras of associative type, it is proved that any module over a semisimple algebra is completely reducible. We also prove that any left homogeneous ideal of a semisimple algebra of associative type is generated by a homogeneous idempotent.     

<Emphasis Type="Italic">n</Emphasis>-Tuple algebras of associative type

We study properties of n-tuple algebras of associative type. We show that the nilpotency of an n-tuple algebra of associative type is determined by the nilpotency of each element. In addition, we characterize the nilpotency of an n-tuple algebra of associative type in terms of the trace function. In the final part of the paper, we show that a homogeneously semisimple n-tuple algebra of associative type is the direct sum of two-sided ideals each of which is a homogeneously simple n-tuple algebra of associative type.     

On the nilpotency and decomposition of Lie-type algebras

In the paper, an analog of the Engel theorem for graded algebras admitting a Lie-type module is proved. Moreover, it is shown that every semisimple algebra of associative type with ordered grading and one-dimensional grading subspaces is the direct sum of two-sided ideals that are simple algebras.     

On structures of modular adjacency algebras of Johnson schemes

In this paper, we consider algebras over a field of characteristic p, which are generated by adjacency algebras of Johnson schemes. If the algebra is semisimple, the structure is the same as that of the well-known Bose-Mesner algebras. We determine the structure of the algebra when it is not semisimple.     

Genetic Volterra algebras and their derivations

The present paper is devoted to genetic Volterra algebras. We first study characters of such algebras. We fully describe associative genetic Volterra algebras, in this case all derivations are trivial. In general setting, i.e., when the algebra is not associative, we provide a su?cient condition to get trivial derivation on generic Volterra algebras. Furthermore, we describe all derivations of three dimensional generic Volterra algebras, which allowed us to prove that any local derivation is a derivation of the algebra.     

Restricted Lie algebras all whose elements are semisimple

People studied the properties and structures of restricted Lie algebras all whose elements are semisimple. It is the main objective of this paper to continue the investigation in order to obtain deeper structure theorems. We obtain some sufficient conditions for the commutativity of restricted Lie algebras, generalize some results of R. Farnsteiner and characterize some properties of a finite-dimensional semisimple restricted Lie algebra all whose elements are semisimple. Moreover, we show that a centralsimple restricted Lie algebra all whose elements are semisimple over a field of characteristic p > 7 is a form of a classical Lie algebra.     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Relations Between Algebras and Their Subalgebras of Invariants

We prove that, if H is a finite-dimensional semisimple Hopf algebra, and A is an FCR H-module algebra over an algebraically closed field, then A is a PI-algebra, provided the subalgebra of invariants is a PI-algebra. We also show that if A is an affine algebra with an action of a finite group G by automorphisms, the subalgebra of the fixed points A^G is in the center of A, and the characteristic of the ground field is either zero or relatively prime to the order of G, then A^G is affine. Analogous results are proved for graded algebras and H-module algebras over a semisimple triangular Hopf algebra over a field of characteristic zero. We prove also that, if A is an H-module algebra with an identity element, and H is either a semisimple group algebra or its dual, then, if A is semiprimitive (semiprime), then so is A^H.     

Hadamard algebras

The existence problem for Hadamard decompositions in semisimple associative finite-dimensional complex algebras is studied. Under the assumption that the well-known hypothesis of the Hadamard matrices is satisfied, this problem is completely solved for algebras isomorphic to the direct sum of a matrix algebra of order 2 and a semisimple commutative algebra.