By Y. Bahturin

ISBN-10: 9048143179

ISBN-13: 9789048143177

ISBN-10: 9401708398

ISBN-13: 9789401708395

This e-book has built from a chain of lectures which have been given via the writer in mechanics-mathematics division of the Moscow country college. In 1981 the direction "Additional chapters in algebra" changed the direction "Gen eral algebra" which used to be based by means of A. G. Kurosh (1908-1971), professor and head of the dept of upper algebra for a interval of a number of a long time. the fabric of this path shaped the root of A. G. Kurosh's recognized booklet "Lectures on basic algebra" (Moscow,1962; 2-nd variation: Moscow, Nauka, 1973) and the e-book "General algebra. Lectures of 1969-1970. " (Moscow, Nauka, 1974). one other booklet according to the path, "Elements of normal al gebra" (M. : Nauka, 1983) was once released through L. A. Skorniakov, professor, now deceased, within the related division. it may be famous undefined. G. Kurosh used to be not just the lecturer for the direction "General algebra" yet he was once additionally the famous chief of the medical university of an analogous identify. it truly is tricky to figure out the bounds of this college; despite the fact that, the "Lectures . . . " of 1962 males tioned above comprise a few fabric which exceed those limits. ultimately this impression intensified: the lectures of the direction got via many recognized scientists, and a few of them see themselves as "general algebraists". each one lecturer introduced major originality not just in presentation of the cloth yet within the substance of the path. accordingly no longer all fabric that's now authorised as invaluable for algebraic scholars suits in the scope of normal algebra.

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**Example text**

1 = xr + m, for some r E R, mE M. Going over to equivalence classes we get 1 + M =(x + M)(r + M), so that r + M =(x + M)-l, and RIM is a field. 0 There is the classical problem in mathematics of extending numerical systems in such a way that algebraic equations can be solved. In order to solve equations of the form mX + n = 0, where m and n are natural numbers, rational numbers are introduced. An equation of the form xn = m requires irrational numbers and the equation X2 + 1 =0 the complex numbers.

3. MODULES AND REPRESENTATIONS 33 A submodule N in M is a subgroup in M such that rx E N for any r E R and any x EN. In this case the quotient group MIN has well defined R-module structure (quotient module) if we set r(x + N) = rx + N. A homomorphism q, : Ml -+ M2 of one R-module into another is a homomorphism of abelian groups such that for any x E Ml and any r E R we have q,(rx) = rq,(x). A bijective homomorphism of modules is called an isomorphism. e. the set of elements with zero image, is an R-submodule.

0 Literature. Kostrikin [1], Lang [1], Kurosh [1]. CHAPTER I COMMUTATIVE ALGEBRA § 1. 1. Concept of finite extension. Let K be a field, F a subfield. We then say that K is an extension of the field F. We can see K as a vector space over F by associating to a scalar A E F and a vector x E K the vector Ax, which is equal to the usual product of elements A, x E K. If the dimension dimF K of this vector space is finite then K is called a finite extension of the field F, and dimFK is called the degree of this extension (often it is denoted as (K : F) or [K : F]).

### Basic Structures of Modern Algebra by Y. Bahturin

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