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

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

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Then r £ R for a suitable regular subset i^ of 5, whence it follows that r is the chosen element ( b y / ) of the set of those ele­ ments of S, which are greater than all the elements of R(r). Since, according to (d)? R(r) = R*(r)9 the above holds for R* instead of R9 whence we have proved that R* is regular. Now, R* has, according to the premise of the theorem, an upper bound s* (in S). If moreover s* is not a maximal element of S9 then there is an 2/a R - A 24 SET-THEORETICAL PRELIMINARIES element x in S which is greater than s*.

Those natural numbers p > 1 which are divisible by no natural number except 1 and pare called prime numbers. A natural number which is neither 1 nor a prime number, is called a composite number. The composite num­ bers are the products ab (a, b = 2, 3, 4 , . . ) . Each composite number is divisible by at least one prime number, and the least among its divisors d ( > 1) is certainly a prime number. The next remark is of fundamental importance. According to G.

Therefore we call the order of a nilpotent element also its degree of nilpotence. The zero element is nilpotent. The rest of the nilpotent elements are zero divisors. Except for the zero element there are no elements simultaneously nilpotent and idempotent. For an a (6 F) and a natural number n we denote by %/OL the solution of the equation |" = a and call it an nth radical of a. According to this, ^fi is in general a manyvalued symbol. If the equation has no solution, we say that ^/a is non-exist­ ent.

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