Tang Gaohua, Su Huadong, Commutative ring and zero-divisor semigroups of regular polyhedrons.
乔晓云，唐高华，郑学谦。单圈图的全图谱半径，广西科学15（2008），No.3，224-227 4。
Let R be a commutative Noetherian local ring, M a finite R-module and I an ideal of R.
设是交换的Noetherian局部环，I是R中的理想，M是有限生成的R-模。
Using properties of zero-divisors, regular elements and subdirectly irreducible ring, we obtain some results on the commutativity of semi-prime ring and Jacobson semi-simple ring as follows 1 Let R be a semi-prime ring, a∈R and 2a is not zero-divisor, If R satisfies one of the conditions in this paper, then R is commutative.
在某些特殊环的交换性方面取得了进一步的结果，并得到了一些新的结论，主要有一设R 为半质环，a∈R ，且2a 为非零因子，如果R 满足论文中八个条件之一，则可以证明环R 为交换环。
Let R be a Jacobson semi-simple ring, a ∈R and 2a is not zero-divisor. If R satisfies condition ∈Z, for any elements x, y in R, which n is an fixxed integer, then R is commutative.
三设R为Jacobson半单纯环，a∈R，且2a为非零因子，若对于任意x，y∈R，有∈Z，n为固定正整数，那么R 为交换环。
We introduce the concept of compatibility of an ordering of higher level with a valuation pair for a commutative ring with unit.
本文的主要目的是把域上高层序与赋值对的相容性推广到含单位元的交换环上来。
Xu Jinzhong proved that a simple module over commutative ring is injective if and only if it is flat.
徐金中证明了交换环上单模是内射的当且仅当它为平坦的。
A general method to show a commutative ring R with 1 to be not universal for GE〓 is presented.
给出了一种证明含1交换环R为非GE〓泛的环的较一般的方法。
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。
Every maximal ideal of every commutative ring with identity is a prime ideal.
带有单位元的交换环的任何极大理想是素理想。
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。
Group, Subgoup, Permutation Group, Cyclic Group, Commutative Group, Normal Subgroup, Factor Group, Isomorphism, Ring, Ideal, Field, Polynomial Ring, Factor Ring.
课程内容：群，子群，置换群，循环群，交换群，正规子群，商群，群之同构，环，理想，体，多项式环，商环等。
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)。
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。
Fermat ring is defined and its some properties are considered. The classification of commutative Fermat ring is determined, and Chevalley-Warning theorem over finite fields is extended to commutative Fermat ring.
定义了抽象Fermat环并研究了它的一些性质，在交换的情形下，完全决定了Fermat环的分类并且证明了数论中著名的Chevalley-Warning定理在交换Fermat环中成立。
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。
If H is infinite dimensional and Φ is a surjective commutative zero-product preserving additive maps on B (H), It is obtained that Φ is a nonzero scalar multiple of a ring isomorphism or a ring anti-isomorphism.
进一步给出了无限维情形下，若Φ是保交换零积可加满射，则Φ是非零数乘一个环同构或一个环反同构。
Let R be a commutative ring with 1 and let 2 be an unit of R.
设R是有1的交换环，2是R的单位。
Compatibility of an ordering of higher level with a valuation pair for a commutative ring
交换环上高层序与赋值对的相容性
The spectrum of a non commutative ring is not a spectral space under Zariski topology, but a spectral closure of the spectrum can be constructed.
非交换环的素谱按Zariski-拓扑不成谱空间，但可以构造它的谱闭包。
The commutative problem of arbitrary PI ring is the main research contents of the commutative theory.
任意PI-环的交换性问题是环的交换性理论的一个主要内容。