Loday introduced di-associative algebras and tri-associative algebras motivated by periodicity phenomena in algebraic K-theory.The purpose of this paper is to study the splittings of operations on di-associative algeb...Loday introduced di-associative algebras and tri-associative algebras motivated by periodicity phenomena in algebraic K-theory.The purpose of this paper is to study the splittings of operations on di-associative algebras and tri-associative algebras.We introduce the notion of a quad-dendriform algebra,which is a splitting of a di-associative algebra.We show that a relative averaging operator on dendriform algebras gives rise to a quad-dendriform algebra.Furthermore,we introduce the notion of six-dendriform algebras,which are splittings of the tri-associative algebras,and demonstrate that homomorphic relative averaging operators induce six-dendriform algebras.展开更多
在环论中,幂等元是很重要的一类元素,幂等元是指满足a2=a的元素a。任何含单位元的环通常都有两个幂等元,即0和1,这两个特殊的幂等元通常被称为平凡幂等元。然而,在环ℤn和ℤn[ x ]中,可能存在非平凡幂等元。本文将研究多项式环ℤp2q2[ x ]...在环论中,幂等元是很重要的一类元素,幂等元是指满足a2=a的元素a。任何含单位元的环通常都有两个幂等元,即0和1,这两个特殊的幂等元通常被称为平凡幂等元。然而,在环ℤn和ℤn[ x ]中,可能存在非平凡幂等元。本文将研究多项式环ℤp2q2[ x ]中的幂等元,并进一步探究多项式环ℤp2q2[ x ]上的2阶矩阵环M2(ℤp2q2[ x ])中非平凡幂等元的形式与性质,其中p、q为不同素数。研究结果表明,ℤp2q2[ x ]中有4个幂等元,M2(ℤp2q2[ x ])中有7个非平凡幂等矩阵。记环R的幂等元集合为Id(R)。展开更多
基金Supported by the Science and Technology Program of Guizhou Province(Grant No.QKHJC QN[2025]362)the National Natural Science Foundation of China(Grant No.12361005).
文摘Loday introduced di-associative algebras and tri-associative algebras motivated by periodicity phenomena in algebraic K-theory.The purpose of this paper is to study the splittings of operations on di-associative algebras and tri-associative algebras.We introduce the notion of a quad-dendriform algebra,which is a splitting of a di-associative algebra.We show that a relative averaging operator on dendriform algebras gives rise to a quad-dendriform algebra.Furthermore,we introduce the notion of six-dendriform algebras,which are splittings of the tri-associative algebras,and demonstrate that homomorphic relative averaging operators induce six-dendriform algebras.
文摘在环论中,幂等元是很重要的一类元素,幂等元是指满足a2=a的元素a。任何含单位元的环通常都有两个幂等元,即0和1,这两个特殊的幂等元通常被称为平凡幂等元。然而,在环ℤn和ℤn[ x ]中,可能存在非平凡幂等元。本文将研究多项式环ℤp2q2[ x ]中的幂等元,并进一步探究多项式环ℤp2q2[ x ]上的2阶矩阵环M2(ℤp2q2[ x ])中非平凡幂等元的形式与性质,其中p、q为不同素数。研究结果表明,ℤp2q2[ x ]中有4个幂等元,M2(ℤp2q2[ x ])中有7个非平凡幂等矩阵。记环R的幂等元集合为Id(R)。
