数学专业英语－Groups and Rings

yunjiang · 发表于 2004-5-6 09:39:03

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>During the present century modern abstract algebra has become more and more important as a tool for research not only in other branches of mathematics but even in other sciences .Many discoveries in abstract algebra itself have been made during the past years and the spirit of algebraic research has definitely tended toward more abstraction and rigor so as to obtain a theory of greatest possible generality. In particular, the concepts of group ,ring,integral domain and field have been emphasized.
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>The notion of an abstract group is fundamental in all sciences ,and it is certainly proper to begin our subject with this concept. Commutative additive groups are made into rings by assuming closure with respect to a second operation having some of the properties of ordinary multiplication. Integral domains and fields are rings restricted in special ways and may be fundamental concepts and their more elementary properties are the basis for modern algebra.
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Groups
 
DEFINITION A non-empty set G of elements a,b,…is said to form a group with respect to 0 if:
I. G is closed with respect to 0
II. The associative law holds in G, that is 
aо(bоc)=(aоb)оc
for every a, b, c of G
Ⅲ. For every a and b of G there exist solutions χ and У in G of the equations
 aοχ=b yοa=b
A group is thus a system consisting of a set of elements and operation ο with respect to which G forms a group. We shall generally designate the entire system by the set G of its elements and shall call G a group. The notation used for the operation is generally unimportant and may be taken in as convenient a way as possible. 
DEFINITION A group G is called commutative or abelian if
aοb=bοa
For every a and b of G.
An elementary physical example of an abelian group is a certain rotation group. We let G consist of the rotations of the spoke of a wheel through multiples of 90º and aοb be the result of the rotation a followed by the rotation b. The reader will easily verify that G forms a group with respect to ο and that aοb=bοa. There is no loss of generality when restrict our attention to multiplicative groups, that is, write ab in stead of aοb.
 
EQUIVALENCE
 
In any study of mathematical systems the concept of equivalence of systems of the same kind always arises. Equivalent systems are logically distinct but we usually can replace any one by any other in a mathematical discussion with no loss of generality. For groups this notion is given by the definition: let G and G´ be groups with respective operations o and o´,and let there be a1-1 correspondence
S : a<v:shapetype> <v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></v:shapetype><v:shape><v:imagedata></v:imagedata></v:shape>a´ (a in G and a´ in G´)
between G and G´ such that
 (aοb)´=a´ο b´ 
for all a, b of G. then we call G and G´equivalent(or simply, isomorphic)groups.
The relation of equivalence is an equivalence relation in the technical sense in the set of all groups. We again emphasize that while equivalent groups may be logically distinct they have identical properties.
The groups G and G´ of the above definition need not be distinct of course and o´ may be o. when this is the case the self-equivalence S of G is called an automorphism. 
 I: a <v:shape><v:imagedata></v:imagedata></v:shape> a
Of G, but other automorphisms may also exist.
 
Rings 
 
A ring is an additive abelian group B such that 
I. the set B is closed with respect to a second operation designated by multiplication; that is , every a and b of B define a unique element ab of B.
II. multiplication is associative; that is 
a (bc) = (ab)c
for every a, b, c of B.
Ⅲ. The distributive laws 
 a (b+c) = ab +ac (b+c) a=ba +ca
hold for every a, b, c of B.
The concept of equivalence again arises. We shall write
 B ≌ B′
to mean that B and B′are equivalent.

yunjiang · 发表于 2004-5-6 09:39:27

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0cm 0cm 0pt; TEXT-INDENT: 21.75pt; TEXT-ALIGN: center" align=center>Vocabulary<

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0cm 0cm 0pt; TEXT-INDENT: 21.75pt">Group 群 rigor 严格ring 环 generalization 推广 integral domain 整环 Abelian group 阿贝尔群commutative additive group 可交换加法群 rotation 旋转automorphism 自同构

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