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Associative Commutative Group Binary Operation Additive Distributive Abelian Group Subtraction

1	Distributive
1	Binary Operation
1	Additive
1	Subtraction
1	Associative
1	Abelian Group
1	Group
1	Commutative

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Ring

A ring is a set together with two binary operations, denoted $+: R \times R \longrightarrow R$ and $\cdot: R \times R \longrightarrow R$ , such that

and $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a,b,c \in R$ (associative law)
for all $a,b \in R$ (commutative law)
There exists an element $0 \in R$ such that for all $a \in R$ (additive identity)
For all $a \in R$ , there exists $b \in R$ such that (additive inverse)
$a\cdot(b+c) = (a \cdot b) + (a \cdot c)$ and $(a+b) \cdot c = (a \cdot c) + (b \cdot c)$ for all $a,b,c \in R$ (distributive law)

Equivalently, a ring is an abelian group

together with a second binary operation $\cdot$ such that $\cdot$ is associative and distributes over

. Additive inverses are unique, and one can define subtraction in any ring using the formula

where

is the additive inverse of

We say has a multiplicative identity if there exists an element $1 \in R$ such that $a \cdot 1 = 1 \cdot a = a$ for all $a \in R$ . Alternatively, one may say that is a ring with unity, a unital ring, or a unitary ring. Oftentimes an author will adopt the convention that all rings have a multiplicative identity. If does have a multiplicative identity, then a multiplicative inverse of an element $a \in R$ is an element $b \in R$ such that $a \cdot b = b \cdot a = 1$ . An element of that has a multiplicative inverse is called a unit of .

A ring is commutative if $a \cdot b = b \cdot a$ for all $a,b \in R$ .