单位元

The operation laws including:commutative law, associative law, cancellation law, unity of operation, inverse element, and distributive law between two operations,here we just omit them.

这些运算律包括：交换律，结合律，消去律，运算单位元，运算逆元，两个运算之间的分配律等，在此不再一一说明详细过程。

If we consider the integers with ordinary multiplication we have closure , associativity, and identity element.

如果我们考虑整数集且用通常的乘法，我们有封闭性，可结合性，和一个单位元。

If R satisfies =0 for any elements x ,y in R then ⑴when k =1, R is commutative;⑵when R has the unity, R is commutative;⑶when R has at least one right regular element and f 2 (t 1,t2) has no terms with degree1 such that Fn= F then R iscommutative when R has at least one right regular element. Here we would like to point out that the above results generalized the results of Fu changlin ,moreover generalized some results of other references . This results enriched the commutative conditions of any ring.

若任取R 中元x， y均有=0 那么⑴k =1时R 为交换环；⑵R 有单位元时R 为交换环；⑶f_2 (t_1，t_2)中t_2的次数不小于k -1且R 中至少有一个右正则元时R 为交换环。

Conversely, if ''S'' is a subgroup of ''G'', then it obeys the axioms of a group.

因此，满足了闭包、单位元和逆元公理，而结合律是继承来的，所以''S''是子群。

A language is called an identity biordered language over A,shortened as IBL,if it is recognized by the identity element of a monoid.

语言称为A上的单位语言，若它被某么半群的单位元所识别。

Using properties of idempotents,we show several necessary and sufficient conditions for the invertibility of the difference of two idempotents, and exhibit some applications to matrix ring s.

本文研究在一个有单位元的环中两个幂等元之差的可逆性问题，利用幂等元的性质，得到了两个幂等元之差可逆的几个充分必要条件，并给出了在矩阵环中的几个应用。

Kyat缅元: Kyat is a basic unit of currency in Burma.

缅元是缅甸的基本货币单位。

The so-called AT-algebras are inductive limits of finite direct sums of matrices over the extension algeras of circle algebra by K, where K is the C~*— algebra of all compact operators on a separable infinite dimensional Hilbert space.

若V_*与V_*同构，且保持单位元等价类；T与T仿射同胚，且同构映射与同胚映射相容，则存在E与E′的同构导出上述同构和同胚，所谓AT-代数即为圆代数通过κ的本质酉扩张的矩阵代数的有限直和的归纳极限，这里κ为可分的无限维复Hilbert空间上的紧算子全体，不变量中的V*为三变元Abel半群，T为迹态空间，[1]为单位元所在的Murray-von Neumann等价类，r_E为连接映射。

We introduce some concepts, such as yon Neumann algebras, factor von Neumann algebras, nest algebras and so on, and give some well-known theorems that we will use in this paper. In Chapter 2, we put our attention on linear maps that preserving zero Jordan triple product on nest subalgebrasof factor yon Neumann algebras.

第二章首先对因子von Neumann代数中套子代数上保Jordan三重零积的线性映射进行了研究，证明了从因子von Neumann代数中套子代数到任一有单位元的Banach代数的保Jordan三重零积的单位线性双射是Jordan同构。

In many people's eyes, real estate agent is dealing with a house, rent to own apartment units sure is easy Journalist in the survey found that many villages in the city are rental agents Farmer's house, 100 yuan cheaper one, the expensive will not exceed 200元, speaking of some brokers, independent kitchen and multi-unit housing is a luxury item.

在很多人看来，房地产经纪人是处理房子，出租给自己的公寓单位务必很容易记者调查发现，许多村庄在城市的出租代理人农民的房子， 100元更便宜的，昂贵的将不超过200元，讲了一些经纪人，独立的厨房和多单元住宅是奢侈品。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失