Get Algebra Volume 1 PDF

By Rédei, L.; Sneddon, I. N.; Stark, M

ISBN-10: 1483197611

ISBN-13: 9781483197616

ISBN-10: 1483222640

ISBN-13: 9781483222646

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Extra info for Algebra Volume 1

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13) for an element a ( ^ 0), then we call a a left zero divisor. ) Of course, zero divisor means an element that is both a left and right zero divisor. By a left annihilator (of F) we mean an element a for which af = 0 (for all f € F). 14) The left annihilators different from 0 are obviously left zero divisors. 15) has at most one solution f. It is important that the definition of left regularity has another formulation. First of all we say: The left-hand cancellation law is valid for an element Q, if from QOL = gj3 it follows that a = /?.

Thus S is always an operator domain for itself in this sense, if a multiplication is defined in S. As mentioned, the application of an operators upon the elements a of S effects a mapping of S into itself, namely the mapping a -> aa. Here we have to point out that the same mapping corresponds to two different operators a, b if aa = 6a for all a. It is evident, conversely, that any mapping may be considered as an operator. ) It should be noted that two different mappings are always different as operators.

This theorem is known in the literature as BIRKHOFF's theorem, however, HAUS­ DORFF (1914) was the first to find it. Cf. also BIRKHOFF (1948). § 15. Theorem of Well-ordering THEOREM 19 (theorem of ZERMELO or the theorem of well-ordering). Any set can be well-ordered. Before the proof we note by way of introduction that: Given two semiordered sets A, B, we shall only then say that A is a subset of B (or B is an o verset of A), if, first of all, the set B contains the set A and any relation x < y (x9y 6 A) which is true in B, is also true in A.

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Algebra Volume 1 by Rédei, L.; Sneddon, I. N.; Stark, M


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