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

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

ISBN-10: 1483222640

ISBN-13: 9781483222646

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Example text

The repeated application of this equation shows that every term of the sequence is equal to one of the first n terms. Since the latter are different, it follows that we have 0(a) = n, which completes the proof. We gather from the proof that the sequence a, a 2 , . . in the case of an a of finite order is always of the form a, . . , a m ; a m+1 , . , a"; a w+1 , . , a " ; . . 22) where a , . , a71 are all different. We can call a , . , a"1 the aperiodic part and a m + 1 ,. , a" the period of the sequence a, a 2 , .

The natural numbers; 7. the rational num­ bers; 8. the positive numbers. (Cf. ) For the sake of clarity we give the above definitions in tabular form as follows group (Abelian group) semigroup module semimodule ring semiring s-field (field) semi-s-field (semifield) X' X +' + +' + +' + X X xw The signs + and X indicate here that addition and multiplication, respectively, are defined in the structure in question; further / indicates that the composition is invertible, and [/] indicates that it is invertible for the set of elements different from 0.

Similarly, we also have to include rings and skew fields among modules and also fields among commutative rings. a. Of these two products one is called the dual of the other. The passing from one product to the dual STRUCTURES 39 product is called dualization. From any concept connected with multipli­ cation, by the dualization of all products another concept is formed, which we call the dual of the former. We can speak in a similar way of dual defi­ nitions and theorems. Those concepts (definitions, theorems) which are invariant under dualization are called self-dual.

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