# Some Related Algebraic Structures

Chapter

## Abstract

Let

*R*be an additive abelian group containing elements 0,*a*,*b*,*c*, …. It is called a*ring*if it is also closed with respect to a second composition called*multiplication*which is both associative and distributive. Thus, the elements of a ring*R*must, in addition to the axioms (1.2.1a) of Section 1.2, also satisfy the following requirements:- (i)
*a*∈*R*;*b*∈*R*⇒*ab*∈*R*for any*a*,*b*. - (ii)
*a*(*bc*) = (*ab*)*c*for any*a*,*b*,*c*∈*R*. - (iii)
*a*(*b*+*c*) =*ab*+*ac*and (*b*+*c*)*a*=*ba*+*ca*.

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## References

- 1.
**A. Kurosh**,*Higher Algebra*, translated from the Russian by George Yankovksy, Moscow, Mir Publishers, 1972.zbMATHGoogle Scholar - 2.
**B.L. Van der Waerden**,*Modern algebra*vols I,II, (In part a development from lectures by E. Artin and E. Noether) New York, F. Ungar, c1950–c1953Google Scholar - 3.

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