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    Ring Theory

    Ring theory originated from Fermat's last theorem and is a subtopic of abstract algebra that studies rings. Rings are defined to be an abelian group that includes both operations of addition and multiplication. A set R is a ring if it satisfies the associative property, distributive property and has a multiplicative identity. The associative property for multiplication would be satisfied if:

    (a x b) x c = a x (b x c)

    holds true for all a, b, and c where a, b, and c are elements of the ring. The distributive property would be satisfied if the following equality were satisfied similarly:

    a x (b + c) = (a x b) + (a x c)


    (b + c) x a = (b x a) + (b x c)

    Furthermore, there must exist an element 'y' such that for all elements of the ring such as 'a' this is true:

    y x a = a x y = a

    This is what is meant by having a multiplicative identity. If only these three properties were satisfied then the ring would be a non-commutative ring. In order for the ring to be a commutative ring it would also have to satisfy the commutative property. That is to say

    a x b = b x a

    holds true for all the elements of the ring. The above formulations are for the operation of multiplication. Similar properties for addition would also have to be satisfied. A common example of some rings are the set of integers, the set of rational numbers, the set of real numbers, and the set of complex numbers.

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