Ring

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Formal Definition

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A ring $\langle R , +, \cdot \rangle$ is a set $R$ together with two binary operations, denoted $+$ and $\cdot$ satisfying three axioms:

(1)

\begin{align} \Re _1 : \langle R , + \rangle \textrm{ is an abelian group} \newline \Re _2 : \textrm{ Multiplication is associative. That is, for all } a,b,c \in R, (a \cdot b) \cdot c = a \cdot (b \cdot c) \newline \Re _3 : \textrm{ For all} a,b,c \in \mathbb{R}, \textrm{ the left distribution law: } a \cdot (b+c) = a \cdot b+a \cdot c \newline \textrm{ and the right distribution law: } (a+b) \cdot c = a \cdot c +b \cdot c \textrm{ hold}. \end{align}

Informal Definition

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A ring may be thought of as a group with an additional operation. This second operation does not follow all of the same rules as the group operation, but is associative. Additionally, the new operation is related to the other operation via the right and left distribution laws.
Example(s)

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$\mathbb{Z }, \mathbb{Q }, \mathbb{R }, \mathbb{C },$ under $\cdot$ and $+$ are rings.
$\langle M_n ( R ), +, \cdot \rangle$ where $M_n$ denotes $n \times n$ matrices, and $R$ denotes any ring.
Defining $F$ as the set of all real-valued functions $f: \mathbb{R} \rightarrow \mathbb{R}, \langle F, +, \ddot \rangle$ defines a ring.
$\langle \mathbb{Z} _n , +_n , \cdot _n \rangle$ is a ring.
$\langle 3\mathbb{Z} , + , \cdot \rangle$ is a ring.
If $\Re _1 , \Re _2 , ..., \Re _n$ are rings, then $\Re _1 \times \Re _2 \times ... \times \Re _n$ is a ring.

Non-example(s)

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$\langle \mathbb{R}, + \rangle$ is not a ring because it has only one operation defined on the given set.

Additional Comments

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1. We often write $ab$ for $a \cdot b$
2. multiplication need not be commutative. If it is we say that the ring is commutative.
3. There is no need for the existence of a multiplicative identity.
If there is such an identity, we say it is a ring with unity.
Usually the multiplicative identity is written as $1$.