2 semester

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The name of the discipline

Rings theory

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Curriculum Specialty

Master Program, first year,

Mathematics

3

Semester

Spring semester

4

Semester credit hours

3

5

Full name of the lecturer

Tikhonov S.V.

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Aims of the study of the discipline

To acquaint students with the basic concepts and methods of the theory of rings, to study a number of the most important classes of rings, and also the most important classical results about rings, t to instill in students the ability to independently study educational and scientific literature in the field of mathematics, to expain some non-trivial constructions and techniques , used by working specialists.

As a result of studying the discipline the student should be able to:

to express a symmetric polynomial elementary symmetric;

– perform actions in the factor ring;

– finding generators of ideals in and out .

– decompose the ring into a straight product of a prick;

– perform basic operations on ideals of rings;

– apply the most important theoretical results to the solution of computational problems related to rings

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Prerequisites

algebra and number theory

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The content of the discipline

A definition of a ring and a subring, examples. The multiplicative group of a ring.

A homomorphism, an isomorphism of rings, the kernel of a homomorphism.

Ideals of rings. Quotient rings.

First Isomorphism Theorem for rings. Principal ideals. Principal ideal rings. Ideals in and .

Maximal ideals and their corresponding quotient rings. Maximal ideals in .

Simple ideals and their corresponding quotient rings.

Direct products of rings. The structure of the ring and the arithmetic corollaries.

Operations on ideals. The radical of an ideal.

Noetherian rings. Hilbert’s basis theorem.

The ring of polynomials in n variables. Monomial orders. The main theorem on symmetric polynomials. Newton’s Formulas.

Quotient rings.

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The recommended literature

  1. Vinberg E.B. A course in algebra. AMS, Graduate Studies in Mathematics, V: 56. 2003; 511 pp.
  2. Kostrikin A.I. Introduction to algebra. Springer-Verlag, 1982. 575 pp.
  3. М. Атья, И. Макдональд. Введение в коммутативную алгебру. М.: Мир, 1972.

Advanced reading:

  1. Lang S. Algebra. Springer-Verlag, 2002. 933 pp.
  2. Van der Waerden B.L.. Algebra, Vol. 1-2. Springer-Verlag, 2003.
  3. Kostrikin A.I. Exercises in algebra: A collection of exercises, in algebra, linear algebra and geometry. CRC Press, 1996. 474 pp.

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Methods of teaching

Verbal, visual, problem-based, practical, dialog-based and heuristic.

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Language of education

Russian

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Conditions (requirements), formative and summative assessment

– check of individual tasks,

– control work,

– test.

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System of assesment

control test