由于域上的典型线性李代数都是单李代数,而交换幺环上的典型线性李代数未必仍为单李代数,而单李代数的结构分类对研究半单纯李代数的结构分类,以及可解、幂零李代数的研究至关重要,在这里我们得出了交换幺环上典型线性李代数为R-单李代数的充要条件。

Because the classical linear Lie algebra over a field is a simple Lie algebra, but the classical linear Lie algebra over a ring with identity may not still be a simple Lie algebra, and the classification and structure of simple Lie algebras is critical to the study of the the structure and classification of the semi-simple Lie algebra, even solvable, nilpotent Lie algebra, here we give the necessary and sufficient condition on which the R-classical linear Lie algebra is R-simple Lie algebra.

综述了域上或交换代数上的线性（Ω-）代数的相应的簇（范畴）的Gr迸bner-Shirshov基理论的新成果，如：结合代数（包括群（半群）代数），自由代数的张量积，李代数，Di-代数，pre-李代数，Rota-Baxter代数，metabelian李代数，L-代数，半环代数，范畴代数，等。其中包含了许多应用，尤其是给出了一些著名结论的新的证明。

Some results are reviewed in Gr?bner-Shirshov bases method for different varieties( categories) of linear (Ω-) algebras over a field k or a commutative algebra K over k: associative algebras ( including group ( semigroup) algebras) , tensor product of free associative algebras, Lie algebras, dialgebras, pre-Lie ( Vinberg right ( left) symmetric) algebras, Rota-Baxter algebras, metabelian Lie algebras, L-algebras, semiring algebras, category algebras, etc.There are some ap-plications particularly to new proofs of some known theorems.

设R是有单位元的交换环,A,B都是R上的酉代数,M是非零(A,B)-酉双模,且作为左A-模和右B-模都是忠实的.记T=(A M0B)为由A,B,M构成的三角代数,D为T的导子.给出T满足[D(X),D(Y)]=0的导子的结构,并证明了三角代数T的导子都不是强保交换的.

Let Rbe a commutative ring with identity,A,B be unital algebras over Rand Mbe a nonzero unital (A,B )-bimodule,which is faithful as a left A-module and also as a right B-module. Let T=(A M 0 B)be a triangular algebra,and D be a derivation of T.We described the structure of derivation D which satisfies [D (X ),D (Y )]= 0.We also proved that derivations on a triangular algebra Twill not be strong commutativity-preserving.

通过对一、二代落叶松人工幼龄林土壤交换性阳离子含量的分析比较,结果表明：一、二代落叶松人工幼龄林土壤均无酸化、碱化趋势;一、二代落叶松幼龄林相比,除根际一代落叶松幼龄林交换性钠略低于二代落叶松幼龄林,土壤深度60cm时,盐基饱和度略低于二代落叶松幼龄林外,在土壤其他深度,一代落叶松幼龄林交换性阳离子含量、盐基总量,交换性阳离子钠、钾、钙、镁的含量均高于二代落叶松幼龄林;落叶松连栽使土壤地力下降,影响树木生长,连栽代数增加,土壤盐基总量、阳离子交换量,交换性钾、镁的含量与落叶松代数关系密切。

The content of soil exchangeable cations for first & second-generation larch plantation were analyzed .Re-sult shows that :soil of first & second-generation larch have no trend of acidification & alkalization ;first-generation larch compared with the second-generation larch ,exchangeable sodium of first-generation larch rhizosphere is slight-ly lower than that of second-generation larch ;while soil depth being 60 cm ,base saturation is slightly lower than that of the second-generation larch plantation ;among other soil depth ,the content of exchangeable cations ,total ex-changeable base(TEB) ,exchangeable cations sodium ,potassium ,calcium and magnesium of first-generation larch are higher than that of the second-generation larch ;continuous planting larch can make soil fertility decline ,affect tree growth ,increase continuous plantation ,TEB ,the content of cations exchange capacity (CEC) ,exchangeable po-tassium & magnesium content of soil are closely related to generation of larch

运用首次积分法和交换代数的环理论分别研究得到了两类推广的Davey-Stewartson和Mikhailov-Shabat方程的实或复的精确行波解.

In this paper ,several general real or complex exact traveling wave solutions of the generalized Dav‐ey‐Stewartson equation and the Mikhailov‐Shabat equation are obtained by using the first integral method and the ring theory of commutative algebra .