Rings and Ideals (Carus Mathematical Monographs 8)

RINGS AND IDEALS......Page 1
Carus Mathematical Monographs......Page 4
Title Page......Page 5
Copyright Page......Page 6
Preface......Page 8
Table of Contents......Page 12
1. Definition of a ring......Page 14
2. Examples of rings......Page 15
3. Properties of addition......Page 23
4. Further fundamental properties......Page 26
5. Division rings and fields......Page 32
6. Equivalence relations......Page 40
References......Page 43
7. Definitions and simple properties......Page 44
8. Division transformation. Factor and remainder theorems......Page 49
9. Polynomials with coefficients in a field......Page 55
10. Rings of polynomials in several indeterminates......Page 59
References......Page 64
11. Ideals......Page 65
12. Ideals generated by a finite number of elements......Page 67
13. Residue class rings......Page 74
14. Homomorphisms and isomorphisms......Page 80
15. Additional remarks on ideals......Page 90
16. Conditions that a residue class ring be a field......Page 92
References......Page 95
17. A fundamental theorem......Page 96
18. Rings without unit element......Page 100
19. Rings of quotients......Page 101
References......Page 107
20. Prime ideals......Page 109
21. The radical of an ideal......Page 111
22. A maximum principle......Page 113
23. Minimal prime ideals belonging to an ideal......Page 116
24. Maximal prime ideals belonging to an ideal......Page 120
References......Page 126
25. Direct sum of two rings......Page 127
26. Direct sum of any set of rings......Page 131
27. Subrings of direct sums and subdirect sums......Page 133
28. A finite case......Page 137
29. Subdirectly irreducible rings......Page 141
30. The Jacobson radical and subdirect sums of fields......Page 145
References......Page 150
31. Algebra of logic and algebra of classes......Page 151
32. Boolean rings......Page 153
33. The p-rings......Page 157
34. Regular rings......Page 160
References......Page 162
35. Introduction......Page 163
36. Definitions and fundamental properties......Page 164
37. Determinants and systems of linear homogeneous equations......Page 168
38. Characteristic ideal and null ideal of a matrix......Page 175
39. Resultants......Page 181
40. Divisors of zero......Page 188
References......Page 192
41. Primary ideals......Page 193
42. The intersection of primary ideals......Page 196
43. The prime ideals belonging to a......Page 200
44. Short representation of an ideal......Page 203
45. Noetherian rings......Page 206
46. Ideals and algebraic manifolds......Page 215
References......Page 222
Bibliography......Page 223
Index of Terms......Page 228
Back Cover......Page 230