topic entry on algebra

The subject of algebra may be defined as the study of algebraicsystems, where an algebraic system consists of a set togetherwith a certain number of operations, which are functions (orpartial functions) on this set. A prototypical example of analgebraic system is the ring of integers, which consists of theset of integers, $\\{\\ldots,-2,-1,0,1,2,\\ldots\\}$ togetherwith the operations $+$ and $\\times$ .

In addition to studying individual systems, algebraists considerclasses of systems defined by common properties. For instance,the example cited above is an example of a ring, which is an algebraicsystem with two operations which satisfy certain axioms, such asdistributivity of one operation over the other.

The reason for considering classes of systemsis in order to save work by stating and proving theorems at theappropriate level of generality. For instance, while the statementthat every integer equals the sum of four squares is specific to thering of integers (there are many rings in which this is not the case)and its proof makes use of specific facts about integers, the proofof the fact that the product of two sums of integers equals the sumof all products of numbers appearing in the first sum by numbersappearing in the second sum only involves the distributive law, soan analogous theorem will hold for any ring. Clearly, it iswasteful to restate the same theorem and its proof for every ringso we state and prove it once as a theorem about rings, then applyit to specific instances of rings.

1.
http://planetmath.org/node/ConceptsInAbstractAlgebraConcepts in abstract algebra
2.
topics on group theory
3.
topics on ring theory
4.
topics on ideal theory
5.
topics on field theory
6.
topics on homological algebra
7.
topics on category theory
8.
algebraic k-theory
9.
Special notations in algebra
10.
Topics on polynomials
11.
Topics on field extensions and Galois theory
12.
Entries on finitely generated ideals
13.
http://planetmath.org/node/2530Topic entry on linear algebra
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http://planetmath.org/node/5663Concepts in linear algebra
15.
Matrices of special form
16.
http://planetmath.org/MatrixFactorizationMatrix decompositions
17.
Bibliography for group theory
18.
topics on universal algebra

单词	TopicEntryOnAlgebra
释义	topic entry on algebra The subject of algebra may be defined as the study of algebraicsystems, where an algebraic system consists of a set togetherwith a certain number of operations, which are functions (orpartial functions) on this set. A prototypical example of analgebraic system is the ring of integers, which consists of theset of integers, $\\{\\ldots,-2,-1,0,1,2,\\ldots\\}$ togetherwith the operations $+$ and $\\times$ . In addition to studying individual systems, algebraists considerclasses of systems defined by common properties. For instance,the example cited above is an example of a ring, which is an algebraicsystem with two operations which satisfy certain axioms, such asdistributivity of one operation over the other. The reason for considering classes of systemsis in order to save work by stating and proving theorems at theappropriate level of generality. For instance, while the statementthat every integer equals the sum of four squares is specific to thering of integers (there are many rings in which this is not the case)and its proof makes use of specific facts about integers, the proofof the fact that the product of two sums of integers equals the sumof all products of numbers appearing in the first sum by numbersappearing in the second sum only involves the distributive law, soan analogous theorem will hold for any ring. Clearly, it iswasteful to restate the same theorem and its proof for every ringso we state and prove it once as a theorem about rings, then applyit to specific instances of rings. 1. http://planetmath.org/node/ConceptsInAbstractAlgebraConcepts in abstract algebra 2. topics on group theory 3. topics on ring theory 4. topics on ideal theory 5. topics on field theory 6. topics on homological algebra 7. topics on category theory 8. algebraic k-theory 9. Special notations in algebra 10. Topics on polynomials 11. Topics on field extensions and Galois theory 12. Entries on finitely generated ideals 13. http://planetmath.org/node/2530Topic entry on linear algebra 14. http://planetmath.org/node/5663Concepts in linear algebra 15. Matrices of special form 16. http://planetmath.org/MatrixFactorizationMatrix decompositions 17. Bibliography for group theory 18. topics on universal algebra
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