By Y. Bahturin
This ebook has constructed from a chain of lectures which have been given via the writer in mechanics-mathematics division of the Moscow nation collage. In 1981 the direction "Additional chapters in algebra" changed the path "Gen eral algebra" which was once based by means of A. G. Kurosh (1908-1971), professor and head of the dept of upper algebra for a interval of numerous many years. the cloth of this path shaped the foundation of A. G. Kurosh's recognized booklet "Lectures on normal algebra" (Moscow,1962; 2-nd version: Moscow, Nauka, 1973) and the e-book "General algebra. Lectures of 1969-1970. " (Moscow, Nauka, 1974). one other ebook in response to the path, "Elements of normal al gebra" (M. : Nauka, 1983) used to be released via L. A. Skorniakov, professor, now deceased, within the comparable division. it may be famous undefined. G. Kurosh used to be not just the lecturer for the path "General algebra" yet he was once additionally the well-known chief of the medical university of an analogous identify. it's tricky to figure out the boundaries of this college; although, the "Lectures . . . " of 1962 males tioned above include a few fabric which exceed those limits. ultimately this impression intensified: the lectures of the direction got through many famous scientists, and a few of them see themselves as "general algebraists". every one lecturer introduced major originality not just in presentation of the fabric yet within the substance of the path. as a result no longer all fabric that's now approved as worthwhile for algebraic scholars matches in the scope of normal algebra.
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Additional info for Basic Structures of Modern Algebra
Further, (I 2 + ID)(a) = a(i 2 + 1) = O. So the minimal polynomial of the operator I is equal to X2 + 1. Therefore D decomposes into the sum of proper subspaces U and V corresponding to the roots i and -i of this polynomial. If a E U then ai = I(a) = ia. According to the corollary, a E C. If b E V then bi =I(b) = -ib. We fix a non-zero j-l E V and let b be an arbitrary element from V. e. bj-l ECandbECj. We have: D=UEBV=CEBCj. The basis of the algebra D on R consists of 1, i,j = 1 . j, and k = ij.
I = R. 4. Divisibility in rings. A non-zero element b from the ring F divides an element a E R from the left. if a = be. for a suitable e E R. Analogously is defined divisibility from the right. If R contains an identity element for the multiplication then an element which is a left and right divisor of the identity element. is called an invertible element. o The set R* of invertible elements of the ring R with identity element is a multiplicative group. 0 Thus to any ring ~ with 1 there is associated the group R* of its invertible elements.
26 INTRODUCTION Every field F has a minimal subfield, Fo, the elements of which are fractions of the fonn (ml)(nl)-I, where m and n are integers and nl =I O. If two elements of such fonn coincide, (m 1)( n 1) -I = (m 11)(n 11) -I, but the fractions !!!. l. are not equal, then mnl - min =I 0 and (mnl - mln)1 = O. e. when for some natural 8 =I 0 we have 1 + ... + 1 = 81 = 0, we -----3 say that the field F has non-zero characteristic. The smallest such 8 is then called the characteristic of the field F.