By Rédei, L.; Sneddon, I. N.; Stark, M
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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. When n = 1, then^/a = a, therefore we usually assume n ^ 2. yet is also called the nth root of a and in the cases n = 2, 3 a square root, or cube root, of a respectively.
The task of algebra is the study of the structures. , as S', but for the sake of convenience, we shall indicate the structure in the same way as the set of its elements. With respect to two structures S, T, we say that they are equal (S = T) if and only if they are equal as sets and also if the compositions defined in them are the same. It must now be made clear that the properties of compositions which are valid in a structure will be treated more fully than the nature of the ele ments of the structure.
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.