Contents: Preface. I. Rings, Integral Domains and Fields: 1. Introduction of Rings. 2. Subrings. 3. ring Homomorphism and Ideal of a Ring. 4. Integral Domains and Fields. 5. Ordered Integral Domain and Subfields. 6. Characteristic of a Ring. 7. The field of quotients of an integral domain. 8. Algebra of ideals. 9. Quotient rings (Rings of residue classes). 10. Euclidean Rings (Euclidean Domains). II. Polynomial Rings: 11. Definition. 12. Divisibility and division algorithm for polynomials over a field. 13. Remainder and factor theorems of a polynomial. 14. Euclidean algorithm and greatest common divisors of polynomials over a field. 15. Unique Factorization domain and theorems for polynomials over a field. 16. Eisenstein criterion and polynomials over the rational field. III. Vector Spaces: 17. Definition. 18. Vector subspace and linear span. 19. Linear dependence and independence. 20. Bases and dimensions. 21. sums and direct sums of subspaces and its dimensions. 22. quotient space and its dimensions. 23. vector space homomorphism and linear map. 24. operations on linear maps. 25. existence of complementary subspaces of a subspace of a finite dimensional vector space. IV. Matrices: 26. linear transformation and their representation as matrices. 27. Types of matrices. 28. Operations on matrices. 29. Determinant of a matrix. 30. Product of determinants. 31. Solution of linear simultaneous equations by the method of determinants. 32. Adjoint and inverse of a matrix. 33. Solutions of linear simultaneous equations by matrix method. 34. Rank of a matrix. 35. Linear equations (Homogeneous and non-homogeneous forms and consistency of systems). V. Linear transformations, rank, nullity and dual spaces: 36. Linear transformations or homomorphism on vector spaces. 37. Range of linear transformation. 38. Rank of a linear transformation, rank-nullity theorem. 39. (a) change of basis. (b) linear transformation as vectors, L (U,V) and dual space. 40. Dual basis bidual – second dual space and natural isomorphism. VI. Eigen values and eigen vectors of matrices: 41. Definition of Eigen values, eigen vectors and cayley Hamilton theorem. 42. Diagonalization of square matrices with distinct eigen values.
This textbook presents a lucid and unified description of Abstract algebra at a level which can be easily understood by the students who posses reasonable mathematical aptitude and abstract reasoning. The book provides an example oriented, less heavily symbolic approach to abstract algebra and the text emphasizes specifics such as rings, integral domains and field, polynomial rings, vector spaces, matrices and linear transformations etc. the book concludes with the coverage of eigen values and eigen vectors of matrices. The text of the book is prepared according to the new pattern of UGC by keeping in view the requirement of the students of several universities and colleges.
This book is written to meet the requirements of undergraduate students of several Universities. The topics included in the text fully cover the entire course prescribed by several Universities. (Jacket)