Rings and ideals / by Neal H. McCoy.

Format:

Book

Author/Creator:

McCoy, Neal Henry, 1905- author.

Series:

Carus mathematical monographs ; Number 8.

Carus Mathematical Monographs ; Number 8

Language:

English

Subjects (All):

Rings (Algebra).

Ideals (Algebra).

Algebra, Abstract.

Physical Description:

1 online resource (xii, 216 pages) : digital, PDF file(s).

Edition:

1st ed.

Other Title:

Rings & Ideals

Place of Publication:

Washington : Mathematical Association of America, 1948.

Language Note:

English

Summary:

The MAA is pleased to re-issue the early Carus Mathematical Monographs in ebook and print-on-demand formats. Readers with an interest in the history of the undergraduate curriculum or the history of a particular field will be rewarded by study of these very clear and approachable little volumes. This monograph presents an introduction to that branch of abstract algebra having to do with the theory of rings, with some emphasis on the role of ideals in the theory. Except for a knowledge of certain fundamental theorems about determinants which is assumed in Chapter VIII, and at one point in Chapter VII, the book is almost entirely self-contained. Of course, the reader must have a certain amount of 'mathematical maturity' in order to understand the illustrative examples, and also to grasp the significance of the abstract approach. However, in so far as formal technique is concerned, little more than the elements of algebra are presupposed.

Contents:

""Front Cover""; ""Rings and Ideals""; ""Copyright Page""; ""Table of Contents""; ""Chapter I. Definitions and Fundamental Properties""; ""1. Definition of a ring""; ""2. Examples of rings""; ""3. Properties of addition""; ""4. Further fundamental properties""; ""5. Division rings and fields""; ""6. Equivalence relations""; ""Chapter II. Polynomial Rings""; ""7. Definitions and simple properties""; ""8. Division transformation. Factor and remainder theorems""; ""9. Polynomials with coefficients in a field""; ""10. Rings of polynomials in several indeterminates""

""Chapter III. Ideals and Homomorphisms""""11. Ideals""; ""12. Ideals generated by a finite number of elements""; ""13. Residue class rings""; ""14. Homomorphisms and isomorphisms""; ""15. Additional remarks on ideals""; ""16. Conditions that a residue class ring be a field""; ""Chapter IV. Some Imbedding Theorems""; ""17. A fundamental theorem""; ""18. Rings without unit element""; ""19. Rings of quotients""; ""Chapter V. Prime Ideals in Commutative Rings""; ""20. Prime ideals""; ""21. The radical of an ideal""; ""22. A maximum principle""; ""23. Minimal prime ideals belonging to an ideal""

""24. Maximal prime ideals belonging to an ideal""""Chapter VI. Direct and Subdirect Sums""; ""25. Direct sum of two rings""; ""26. Direct sum of any set of rings""; ""27. Subrings of direct sums and subdirect sums""; ""28. A finite case""; ""29. Subdirectly irreducible rings""; ""30. The Jacobson radical and subdirect sums of fields""; ""Chapter VII. Boolean Rings and Some Generalizations""; ""31. Algebra of logic and algebra of classes""; ""32. Boolean rings""; ""33. The p-rings""; ""34. Regular rings""; ""Chapter VIII. Rings of Matrices""; ""35. Introduction""

""36. Definitions and fundamental properties""""37. Determinants and systems of linear homogeneous equations""; ""38. Characteristic ideal and null ideal of a matrix""; ""39. Resultants""; ""40. Divisors of zero""; ""Chapter IX. Further Theory of Ideals in Commutative Rings""; ""41. Primary ideals""; ""42. The intersection of primary ideals""; ""43. The prime ideals belonging to α""; ""44. Short representation of an ideal""; ""45. Noetherian rings""; ""46. Ideals and algebraic manifolds""; ""Bibliography""; ""Index""

Notes:

Title from publisher's bibliographic system (viewed on 02 Oct 2015).

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-61444-008-5