Mathematics Course 111: Algebra I Part III: Rings, Polynomials and Number Theory
نویسنده
چکیده
7 Rings Definition. A ring consists of a set R on which are defined operations of addition and multiplication satisfying the following axioms: • x+ y = y + x for all elements x and y of R (i.e., addition is commutative); • (x+ y) + z = x+ (y + z) for all elements x, y and z of R (i.e., addition is associative); • there exists an an element 0 of R (known as the zero element) with the property that x+0 = x for all elements x of R; • given any element x of R, there exists an element −x of R with the property that x+(−x) = 0; • x(yz) = (xy)z for all elements x, y and z of R (i.e., multiplication is associative); • x(y + z) = xy + xz and (x + y)z = xz + yz for all elements x, y and z of R (the Distributive Law). The first four of these axioms (the axioms that involve only the operation of addition) can be summarized in the statement that a ring is an Abelian group (i.e., a commutative group) with respect to the operation of addition. Example. The set Z of integers is a ring with the usual operations of addition and multiplication. Example. The set Q of rational numbers is a ring with the usual operations of addition and multiplication. Example. The set R of real numbers is a ring with the usual operations of addition and multiplication. Example. The set C of complex numbers is a ring with the usual operations of addition and multiplication. Example. The set Z[x] of all polynomials with integer coefficients is a ring with the usual operations of addition and multiplication of polynomials. Example. The set Q[x] of all polynomials with rational coefficients is a ring with the usual operations of addition and multiplication of polynomials.
منابع مشابه
Algebra Oral Qualifying Exam Syllabus
algebra is the study of fundamental algebraic structures that occur throughout mathematics: groups, rings, modules, categories, and fields—as well as maps between them. A solid grasp of algebra can be useful to students of all mathematical stripes, and accordingly the suggested syllabus is a “strong core plus more” model: a student preparing for an oral qualifying exam in algebra should (1) fee...
متن کاملResults on Generalization of Burch’s Inequality and the Depth of Rees Algebra and Associated Graded Rings of an Ideal with Respect to a Cohen-Macaulay Module
Let be a local Cohen-Macaulay ring with infinite residue field, an Cohen - Macaulay module and an ideal of Consider and , respectively, the Rees Algebra and associated graded ring of , and denote by the analytic spread of Burch’s inequality says that and equality holds if is Cohen-Macaulay. Thus, in that case one can compute the depth of associated graded ring of as In this paper we ...
متن کاملSet Theory and Algebra in Computer Science A Gentle Introduction to Mathematical Modeling
Algebra greatly broadened the very notion of algebra in two ways. First, the traditional numerical domains such as Z, Q R, and C, were now seen as instances of more general concepts of equationally-defined algebraic structure, which did not depend on any particular representation for their elements, but only on abstract sets of elements, operations on such elements, and equational properties sa...
متن کاملAbstract Algebra, 3rd Edition PDF
Algebra, 3rd Edition PDF Widely acclaimed algebra text. This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics. The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the reader's ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2005