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تحميل الدليل التدريبي

أسئلة شائعة


 

 

الحلقات والحقول

الحلقة وزمرة وحداتها وزمرة تماثلاتها الذاتية، المثاليات وحلقات القسمة، الحلقة الرئيسة، المثاليات الأولية والأعظمية، حقل القواسم لحلقة تامة، مميز الحلقة، المجموع المباشر للحلقات، الفضاءات الحلقية، الحلقات الإقليدية، حلقة كثيرات الحدود، جذور كثيرات الحدود على حقل، امتداد الحقول، الامتدادات البسيطة والمنتهية للحقول، الإغلاق الجبري لحقل، حقول الانشطار،الحقول المنتهية.

المتطلب: 343 ريض

 

 

 

SYLLABUS OF 344 MATM ( RINGS AND FIELDS)

CONTENTS

CHAPTER ONE

1.1 DEFINITION AND EXAMPLES OF RING

2.1 SPECIFIED TYPES OF RINGS

3.1 DEFINITION OF A UNIT AND THE GROUP OF UNITS

4.1 DEFINITION OF ZERO DIVISORS AND NON ZERO DIVISORS 

5.1 DEFINITION OF INTEGRAL DOMAIN , DIVISION RING AND FIELD

6.1 THE TEN TYPES OF RINGS  

7.1 THE RELATIONSHIP BETWEEN THE TEN TYPES OF RINGS

8.1 IDEMPOTENT AND NILPOTENT ELEMENTS OF A RING  

9.1 BOOLEAN GING

10.1 SUBRING AND THE CHARACTERISTIC OF THE RING  

 

CHAPTER TWO

1.2 DEFINITION OF IDEALS AND EXAMPLES

2.2 CONSTRUCT NEW IDEALS OF GIVEN ONES

3.2 DEFINITION OF FINITELY GENERATED IDEAL AND THE PRINCIPALIDEAL

4.2 DEFINITION OF CERTAIN BINARY OPERATIONS  ON THE IDEALS OF A RING

5.2 DEFINITION OF THE INTERNAL DIRECT SUM

6.2 FEFINITION OF RING HOMOMORPHISM AND EXAMPLES

7.2 SOME PROPERTIES OF RING HOMOMORPHISM

8.3 DEFINITION OF KERNEL AND IMAGEOF THE RING HOMOMORPHISM AND SOME APPLICATIONS

8.2 DEFINITION OF AN IMBEDDED RING IN ANOTHER RING

 9.2 DEFINITION OF AN EXTERNAL DIRECT SUM

10.2 RELATIONSHIP BETWEEN THE INTERNAL AND EXTERNAL DIRECT  SUM

 

CHAPTER THREE  

1.3 QUOTIENT GING

2.3 FACTORIZATION OF HOMOMORPHISMS

3.3 CLASSICAL ISOMORPHISM THEOREMS

3.3.A  FIRST ISOMORPHISM THEOREM

3.3 B  SECOND ISOMORPHISM THEOREM

3.3 C THIRD ISOMORPHISM THEOREM

 

CHAPTER FOUR

INTEGRAL DOMAIN AND FIELDS

1.4 A.  EVERY FINITE INTEGRAL DOMAIN IS A FIELD

1.4 B. EVERY INTEGRAL DOMAIN WITH ONLY  A FINITE NUMBER

OF IDEALS IS A FIELD

2.4 DEFINITION OF A SUBFIELD

3.4 THE RELATION BETWEEN A SUBRING OF A FIELD AND THE SUBFIELD GENERATED BY THE SUBRING

4.4 CONSTRUCT THE QUOTIENT FIELD OF AN INTEGRAL DOMAIN

 

CHAPTER FIVE

 

MAXIMAL AND PRIME IDEALS

CHAPTER SIX

DIVISIBILITY THEORY IN INTEGRAL DOMAIN

 

1.6  DEFINITION OF UNIQUE FACTORIZATION DOMAIN

2.6 EVERY PRINCIPAL IDEAL DOMAIN IS A UNIQUE FACTORIZATION DOMAIN

3.6 DEFINITION OF AN EUCLIDEAN DOMAIN

4.6 SOME PROPERTIES OF AN EUCLIDEAN DOMAIN

5.6 EVERY EUCLIDEAN DOMAIN  IS A PRINCIPAL IDEAL DOMAIN

 

CHAPTER SEVEN

POLYNOMIAL RINGS

 

1.7 ROOTS OF POLYNOMIALS OVER A FIELD

2.7 DEFINITION OF A PRIMITIVE POLYNOMIAL

3.7 GAUSS, S LEMMA

4.7 THE EISENSTEIN CRITERION

 

CHAPTER EIGHT

 

1.8 FIELD EXTENTIONS

2.8 FINITE AND SIMPLE  EXTENTION  OF  FIELDS

3.8 ALGEBRIC CLOSURE OF A FIELD

4.8 SPLITTING FIELDS

5.8 FINITE FIELDS

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