Appropriate for undergraduate courses, this second edition has a new chapter on lattice theory, many revisions, new solved problems and additional exercises in the chapters on group theory, boolean algebra and matrix theory. The text offers a systematic, well-planned, and elegant treatment of the main themes in abstract algebra. It begins with the fundamentals of set theory, basic algebraic structures such as groups and rings, and special classes of rings and domains, and then progresses to extension theory, vector space theory and finally the matrix theory. The boolean algebra by virtue of its relation to abstract algebra also finds a proper place in the development of the text. The students develop an understanding of all the essential results such as the Cayley's theorem, the Lagrange's theorem, and the Isomorphism theorem, in a rigorous and precise manner. Sufficient numbers of examples have been worked out in each chapter so that the students can grasp the concepts, the ideas, and the results of structure of algebraic objects in a comprehensive way. The chapter-end exercises are designed to enhance the student's ability to further explore and inter-connect various essential notions.
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EXTENSION THEORY 141153
LATTICE THEORY 154166
VECTOR SPACE THEORY 195244
MATRK THEORY 245348
a a b a e G A U B abelian group addition and multiplication atom bijective binary operations Boolean algebra Boolean function called Clearly closure property column commutative ring complement cosets countable defined by f(x definition determinant diagonal disjunctive normal form distributive property divisor of zero eigenvalues eigenvectors equation Example exists factor finite group function f given matrix group G Hence f Hence the result homomorphism implies infinite inner product space integral domain inverse isomorphism lattice left cosets Let G linear mapping linearly independent minimal polynomial non-empty set non-singular matrix non-zero element normal subgroup Note observe orthogonal orthogonal matrix permutation poset positive integer prime Problem Proof Let prove rank real numbers Show Similarly skew-symmetric Solution Let square matrix subgroup of G subring subset subspace surjective symmetric unique vector space x p y
Page 10 - The cartesian product A x В of two sets A and В is defined as the set of all ordered pairs (x, y), where xe A and ye В.