要讨论环,首先假设我们已经对群的相关概念有了解,因为环是建立在一个叫加群的群的概念之上的.所谓加群,就是一个交换群,并且它的代数运算叫做加法,我们用+来表示.很容易可以理解,加群的单位元就是零元而任意一个元素的逆元正是其负元.比如我们常见的整数集合就是一个加群.如果一个加群还存在一个叫乘法的代数运算,并且对于这个乘法运算是封闭的,满足结合律的以及对两个分配率,那么这个加群就是一个环.环对于加法满足交换律而对于乘法却不一定.如果一个环对乘法满足交换律,那么它是一个交换环.群里一定有单位元,而环不一定有单位元.在有的环里还存在一种特殊的元素,两个这种元素的乘积等于零,而他们又都不等于零,那么这种元素就是零因子.剩余类环(在剩余加群的基础上定义了乘法的环)就是含有零因子的环.有一种环叫做整环,假如它对乘法满足交换律,它含有单位元,没有零因子.可以看到,整数环是一个整环.因为环里不一定有单位元,自然也不一定有逆元,(其实,即使有单位元也不一定就有逆元,比如整数环).我们把一个环叫做除环,如果它至少包含有一个不等于零的元,它有单位元,除零元外每个元都有逆元.有理数环就是一个除环(其实它已经是一个域了).进一步分析除环,一个除环的非零子集对于乘法也是封闭的,有单位元又有逆元,所以这个子集对乘法来说做成了一个群,这个群叫乘群,因此不难理解,一个除环实际上是由一个加群和一个乘群凑合而成的.

最后我们来说明这样一个命题:一个至少有两个元而且没有零因子的有限换是一个除环.

取环的非零子集,因为没有零因子,所以它满足消去律,又因为它是有限的,这样就做成一个乘群,所以一个至少有两个元而且没有零因子的有限换是一个除环.

  上午接到舅舅的电话,说现在在长沙参加一个会议,问我能不能和他见见面,我欣然答应了.他乡遇亲人,这是多么有意思的事情啊.

吃完了午饭,舅舅提出要带我去逛逛长沙,说真的,来了这么久,我去过的名胜还只有岳麓山一处,而且还是逃票进去的.我们虽然只有一个下午的时间,但是因为有车,所以我才有幸瞻仰到岳麓书院的古训感受到马王堆的沧桑.走进湖南马王堆博物馆,时间好象就在那一刻凝结起来,虽然只是隔着窗子观看,一件一件文物还是把我们带到了两前多年前那个神秘的遥不可及的年代.我不禁要问,这一匹匹华丽的丝绸是出自妙龄少女的纤纤细手还是垂暮老妪干瘪的手,是贵夫人消磨时光的玩意,还是女奴为了求生的辛苦劳作.那一件件木犊上讲的到底是什么,是四书五经,还是兵家常识,或者就是记录马忘家的奢侈生活的场景.这个坟墓是为辛追夫人修建的,马王把世间所有最美的东西都带给了他的母亲.当年棺木出土的时候,辛追夫人的遗体没有腐烂的现象,对遗体解剖后,竟然发现她"健康状况"良好,肾脏与器官保留完整,最近甚至有报道说,运用现代X光技术测试,老太太的骨质比现在60岁的正常人还好.这的确是一个奇迹.因为有奇迹所以使得这一切更加神秘.现代科学不能解释,我倒希望那里的一切永远是个迷.....

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Yesterday I accompanied David to renew a flash memery disk which had been bought a few days before and turned out to be down just the day before.Our destination was Guo chu IT market,which is about 1 hour's bus away from our campus.It was really an agonizing moment for David because he had to suffer strong bus sickness during the trip.To avoid that,we especially sat next to a window the only opened window in an air conditioning bus,and I kept talkng to him so that he could forget about bitterness .All my efforts turned out to be inefficient when I found him falling silent and having a pale,tough  look on his face.He seemed to be going to vomit... So we had to disembarked several stops before our destination.

After everything was done,we encountered another challenge~go back by bus.He strongly dennied going by bus again.He'd rather walk than sit in a cramped and stuffy compartment again.

Two hours after I was back by bus,he turned up in front me and proclaimed that walking is much more pleasant.

To me a wireless mouse is quite new but far from comfortable. The wireless mouse I just quitted today is a compliment from Toshiba because I bought a laptop of that brand. It's small and shrewd and red colored so it looks cool and fashionable.But the appearances just don't work so well as ordinary looking ones.You can never imagine how fustrated I was only to find the mouse paused all of a sudden when i was just fill with joy playing computer games tomorrow evening.And I tried over and over again,the point on screen did not move any further.Today I rushed to the shop where I gained this mouse to make a complaint and urge them to change it for a better one.After several minutes' waiting,they gave me a brand new wirless mouse looking exactly like the one which had been down the night before. They plugged it and i saw it was working perfectly.To my satisfaction, I said "thank u".When I was back ,I tried it on,I was shocked to realise it was down again!!!Where the problem lies??? So I've made a decision to quit using it. Tomorrow I'll buy a new one.BUT NEVER A WIRELESS MOUSE!!!!!

Cantonese (or Yue) is one of the five major Chinese languages. These are often called "dialects", but in actuality their differences are great enough to consider them separate languages.

Cantonese is spoken by about 100 million people in the southern provinces of Guangdong and Guangxi and in neighboring areas such as Hong Kong and Macao, as well as throughout South-East Asia in such places as Singapore, Malaysia, Thailand and Vietnam. Due to the migration of Cantonese speakers from Hong Kong and the Guangdong area, Cantonese is the dominant form of Chinese spoken in the Chinatowns of many major cities in the United States, Canada, Australia and elsewhere.

The word Cantonese comes from Canton, the former English name of Guangzhou, capital of Guangdong, which was once considered the home of the purest form of Cantonese. However, through years of mass media and pop culture influence, Hong Kong has now truly become the cultural centre of Cantonese.

Although Mandarin (or putonghua) is the standard and official language in mainland China and Taiwan, it has only been around for about 700 or 800 years, compared to the 2000-year history of Cantonese. Cantonese, not Mandarin, is the dominant language in overseas Chinese communities. This comes from the fact that, around the world, the largest flow of Chinese immigrants originates from Hong Kong.

Cantonese is mainly an oral language. People in Hong Kong use standard Chinese (putonghua) when they read and write. They speak Cantonese in their daily interactions with people. As a colloquial language, Cantonese is full of slang and non-standard usage. The language of youth is rapidly evolving, and new slang and trendy expressions are constantly emerging.

The standard written language in Hong Kong is essentially the same Chinese as everywhere else in China. The only difference is that Hong Kong and overseas communities, like Taiwan, have kept what are called traditional characters, whereas mainland China uses simplified characters. In an attempt to increase litteracy in China, thousands of characters were "simplified" in a 1950 spelling reform initiated by chairman Mao Zedong.

It was really a painstaking job to carry my old dilapidated computer to the railway station where it got packed and mailed.Again i experinced the dishonest and wickedness and greed of the people working for the railway.

Finally I got my new PC, a portable PC to be specific.It feels nice ,i especially like the keyboard here.It's TOSHIBA,a real Japanese brand,maybe the only defect of my new machine.I hope to make full use of it ,at least i'll master MATLAB and PHOTOSHOP.I'm sure i can learn more with the help this laptop.

大熱的天我們擠上暸火車。車廂裏的人實在是太多暸，又因為我們隻是去株洲轉車，所以我們站在車廂連接處。車廂裏已經沒有立足之地暸，可還是有人拼命的往上擠，倖虧有個猛男把上他們都踹暸下去，並且關上暸車門，我們纔擺脫暸被擠成肉餅的阨運。40分鍾后我們到暸株洲火車站。十分順利的簽到去漣源的票，就急匆匆的上火車。這次雖然有座位，且又是靠窗的，但是那車太慢，又好像是公交車，縫站必停，所以我們到翔哥傢已經是晚上7點多暸。

他媽媽是村裏的醫生開暸間診所，騰騰和心心睡病房，我和KK睡産房，被蚊子折騰暸一晚。。。

翔哥傢的厠所是在是恐怖，到處是骯髒的糞便和足足有拇指般大小的蒼蠅，他竟然告訴我厠所是不用衝洗的！

金蓮村是一個以工業為主要産業的村子，這完全齣乎我的意料，和我想象中的村莊完全是兩個樣子。那裏地上，牆上，山上，樹上，到處是煤灰。我都不敢大口喘氣，生怕喫進一口煤，就這樣鼻孔裏也都是煤灰。

交通也很不方便。我們為暸去附近的一個水庫，竟坐暸3趟車，而且一趟比一趟擁擠，車廂又特別矮，我隻能低着頭，站着。我實在是要吐暸，不得不下來。。。