By Rédei, L.; Sneddon, I. N.; Stark, M
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Extra info for Algebra Volume 1
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.
Algebra Volume 1 by Rédei, L.; Sneddon, I. N.; Stark, M