ring (notes) by prof m dabeer mughal

Ring (Notes)

by

Prof. M. Dabeer Mughal

Federal Directorate of Education

Islamabad, PAKISTAN

Partial Contents

1. Rings, commutative ring, ring with unity (identity), examples 1

2. Consequences from the definition 3

3. Unit element of ring 4

4. Division ring or skew field, field, zero divisor 5

5. Integral domain 7

6. Characteristic of ring, Boolean ring 8

7. Sub-ring and related theorem 9

8. Centre of ring and related theorem 11

9. Ring homomorphism 16

10. Isomorphism, kernel of φ and related theorems 17

11. Ideal, left ideal 20

12. Right ideal, two sided ideal, examples and theorems 21

13. Principal ideal, principal ideal ring and related theorem 28

14. Quotient ring and related theorem 37

15. Ist fundamental theorem 42

16. Maximal ideal and related theorem 44

17. Prime ideal and related theorem 49

18. The field of quotients of an integral domain 52

19. Prime field 63

Available at http://www.MathCity.org/mscWe are very thankful to Prof. M. Dabeer Mughal for sharing these notes.

MathCity.org is a non-profit organization, working to promote mathematics in Pakistan. Ifyou have anything (notes, model paper, old paper etc.) to share with other peoples, you cansend us to publish on MathCity.org.For more information visit: http://www.MathCity.org/participate

Dated: September 15, 2010

Ring (Notes)

Available at http://www.MathCity.org/msc

Prof. M. Dabeer Mughal

Federal Directorate of Education

Islamabad, PAKISTAN

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Prof. M. Dabeer MughalFederal Directorate of Education

Islamabad, PAKISTAN

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Prof. M. Dabeer MughalFederal Directorate of EducationIslamabad, PAKISTAN

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Prof. M. Dabeer MughalFederal Directorate of Education

Islamabad, PAKISTAN

Available at http://www.MathCity.org/msc

Prof. M. Dabeer Mughal

Federal Directorate of Education

Islamabad, PAKISTANBY:{

Available at http://www.MathCity.org/msc

Prof. M. Dabeer Mughal

Federal Directorate of EducationIslamabad, PAKISTAN

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Prof. M. Dabeer Mughal

Federal Directorate of Education

Islamabad, PAKISTANBY:

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Prof. M. Dabeer MughalFederal Directorate of Education

Islamabad, PAKISTAN

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Prof.

M.

Dabeer

Mughal

Federal

Directorate

ofEducation

Islamabad,PAKISTAN

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc

Available at http://www.MathCity.org/msc