Ring (Notes) by Prof. M. Dabeer Mughal

A handwritten notes of Ring (Algebra) by Prof. M. Dabeer Mughal (Federal Directorate of Education, Islamabad, PAKISTAN). Best to prepare a “Rings & Vector Spaces” section of Algebra paper in MSc (Mathematics).

CONTENTS OR SUMMARY:

• Rings, commutative ring, ring with unity (identity), examples
• Consequences from the definition
• Unit element of ring
• Division ring or skew field, field, zero divisor
• Integral domain
• Characteristic of ring, Boolean ring
• Sub-ring and related theorem
• Centre of ring and related theorem
• Ring homomorphism
• Isomorphism, kernel of $\varphi$ and related theorems
• Ideal, left ideal
• Right ideal, two sided ideal, examples and theorems
• Principal ideal, principal ideal ring and related theorem
• Quotient ring and related theorem
• Ist fundamental theorem
• Maximal ideal and related theorem
• Prime ideal and related theorem
• The field of quotients of an integral domain
• Prime field