Ring

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A ring is a structure of abstract algebra, similar to a group or a field. A ring $R$ is a set of elements with two operations, usually called multiplication and addition and denoted $\cdot$ and $+$ , which have the following properties:

Under the operation +, the ring is an abelian group and so obeys all the group axioms (existence of an identity, existence of inverses, associativity) as well as commutivity.

There exists an element, usually denoted 1, such that $1 \cdot a = a \cdot 1 = a$ for all $a\in R$ . (Multiplicative identity.)

For every three elements $a, b, c\in R$ we have $a\cdot(b\cdot c) = (a\cdot b)\cdot c$ . (Associativity.)

For every three elements $a, b, c\in R$ we have $a\cdot(b + c) = (a\cdot b) + (a\cdot c)$ and $(a + b)\cdot c = (a\cdot c) + (b\cdot c)$ . (Distributivity of multiplication over addition.)

Note especially that multiplicative inverses need not exist, and that multiplication need not be commutative.

Common examples of rings include the integers or the integers taken modulo $n$ , with addition and multiplication as usual. In addition, every field is a ring.

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Ring

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