Put forward a commutative law in the weak fuzzy implication algebra; The properties of weak FI-algebra and FI-algebra are studied.
引入可交换弱FI代数的概念，讨论了弱FI代数和FI代数的几个性质；进一步得到了交换弱FI代数和交换FI代数的一些结果。
In the chapter two, we will recall some basic concepts and properties in commutative algebra and homological algebra.
在第二章中，作者介绍了交换代数和同调代数的一些基本概念和性质。
For a given strictly proper rational function matrix over a commutative Banach algebra, the controllable, observable realizations have been constructed, the uniqueness of minimal realizations under the system equivalence has been proved, also, the necessary and sufficient conditions for decoupling have been given.
对给定的定义在交换Banach代数上的严格真有理函数矩阵，构造了可控型、可观型的实现，证明了其极小实现在系统等价意义下的唯一性；给出了交换B.fi。
At last, it`s proved that the equivalent class of sharp element is sharp element in the quotient algebra of a weak commutative pseudoeffect algebra gotten by normal Riesz ideals.
讨论了弱可换的伪效应代数商代数中可精确测量元与正规Riesz理想之间的关系。
It is proved that a semiring is complemented if and only if it is a direct product of some Boolean algebra and Boolean ring by studying the relations of congruences on the complemented semiring, so the complemented semiring is the commutative semiring.
s-内射模。通过研究可补半环S的性质及S上的同余关系，证明出可补半环与布尔环和布尔代数的直积是等价的，因而可补半环是交换半环。
Let R be an arbitrary commutative ring with identity, gl the general linear lie algebra over R consisting of all n × n matricesover R and with the bracket operation = xy -yx, t the lie subalgebraof gl consisting of all n×n upper triangular (resp., strictly upper triangular) matrices over R and d the lie subalgebra of gl consisting of all n×n diagonal matrices over R.
在第三章中，对R是交换环的情形，讨论了典型李代数的导子代数的结构问题：设R是一个含幺交换环，gl是R上一般线性李代数。t是gl的所有n阶上三角矩阵(相应地，严格上三角矩阵)构成的子代数，d是gl的所有n阶对角阵构成的李代数。
As a branch of modern algebra, finite fields is an abstract system in algebra which satisfies operational laws such as associative law, commutative law, distributive law etc.
作为近世代数学的一个分支，有限域是一个拥有传统算术四则运算的抽象代数系统，它满足结合律、交换律、分配律、消去律等运算法则。
We prove that the Jordan semi-triple derivable maps of the algebra of all n×n matrices over a 2-torsion free commutative ring with unity is the sum of an inner derivation and A_φ, where A_φis the image of A underφapplied entrywise andφis an additive derivation over R.
证明了作用在一个包含单位元的可交换的2-无挠环上的矩阵代数上的拟三重Jordan可导映射是一个内导子与A_φ的和(这里A_φ是A在φ下逐点作用的像)。
For commutative algebra, algebraic geometry, and singularity theory, the SINGULAR computer algebra system provides a large variety of algorithms in the package kernel as well as shared libraries.
对于交换代数、代数几何和奇点理论，SINGULAR计算机代数系统在软件包内核以及共享库中提供了大量的算法。
Ideal is one of the most important concepts in abstract algebra, especially the theory of commutative ring. It is also an indispensable basic tool to study algebraic number theory.
理想概念是现代抽象代数学特别是交换环理论最核心的概念之一，也是研究代数数论学科必不可少的基本工具。
Students should have some familiarity with commutative algebra and basic topology.
学生需要对交换代数和基本拓扑学有一定的了解。
Rules can also be used for commutative and noncommutative algebra.
规则可以用于交换和非交换代数。
Computations in Commutative Algebra (CoCoA) is another free computer algebra system for working with very large integers, rational numbers, and polynomials.
ComputationsinCommutativeAlgebra（CoCoA）是另一个免费计算机代数系统，用于处理超大型整数、有理数和多项式。
Let H be a finite Hopf algebra, B a commutative ring, H 0 the category of all commutative, co-commutative finite algebra, C the category of commutative rings and A of commutative groups.
设H为有限Hopf代数，B为交换环，H0为交换、余交换的有限Hopf代数范畴，C为交换环范畴，A为交换群范畴。
Let H be a finite dimensional Hopf algebra and A a commutative H-module algebra.
设H是有限维Hopf代数，A是交换的H-模代数。
In the paper, we define the associative BCI-algebras with operation ∧, and prove that its BCK-part is commutative BCK-algebra. The following problems are solved: proper class problem, proper subclass problem and cardinal number problem.
本文定义了∧结合bck-代数，解决了真类问题、真子类问题和基数问题，并得到了它的bck-部分是可换的bck-代数。
The realization theory of linear systems defined over a commutative Banach algebra W is studied. For the given strictly proper rational matrix function over W, using the matrix fraction description, the controllable and observable realizations have been constructed.
研究变换Banach代数W上线性系统的实现理论，对给定的W上的严格真有理矩阵函数，利用其矩阵分式描述，构造了其可控形和可观测形的实现；
Real Coefficient Quadratic Equations Formula of Root on Real Commutative Algebra
交换代数中二次方程的求根公式
Discontinuous Characteristic Boundary Value Problems for Hyperbolic Equations in Commutative Quaternion Algebra
交换四元数代数中一类双曲型方程的间断特征边值
In this paper the construction of commutative algebras for a matrix is established according to its Jordan canonical matrix. From this the symmetry algebra of a bilinear system is derived and a way of finding the symmetry group for a nonlinear system is presented.
本文根据矩阵的Jordan标准形建立矩阵的交换代数的构造方法，在此基础上导出双线性系统的对称代数，并给出寻找一般非线性系统的对称群的一种途径。