Ring (mathematical concept)

A ring is a set closed under two binary operations, with some special properties by definition.

Formal definition

Let $R:|R| \ge 1$ be a set, and $+_R,\cdot_R$ two binary operations defined over $R$.

If $\forall a,b,c \in R$:

• $a+_Rb=b+_Ra$ (additive commutativity)
• $(a+_Rb)+_Rc=a+_R(b+_Rc);(a \cdot_R b)\cdot_R c = a \cdot (b \cdot_R c)$ (associativity)
• $\exists 0_R \in R: a+_R0_R=a$ (additive identity)
• $\exists (-a) \in R: a +_R (-a)=0_R$ (additive inverse)
• $a\cdot_R (b+c)=a\cdot_R b +_R a\cdot_R c$ (distributivity in 1st slot)
• $(a+b)\cdot_R c=a\cdot c +_R b\cdot c$ (distributivity in 2nd slot)

Then $(R,+_R,\cdot_R)$ is a ring.

$+_R,\cdot_R$ do not have to take the usual definitions of $+,\cdot $.

The smallest ring is $(\{0_R\},+_R,\cdot_R)$.

Special rings

Commutative ring

If ring $R$ satisfies $\forall a,b \in R: a \cdot b = b \cdot a$ (multiplicative commutativity), then $R$ is commutative.

Ring with unity

If ring $R$ satisfies $\forall a \in R: \exists 1_R, a\cdot 1_R=a=1_R \cdot a$ (multiplicative identity), then $R$ is a ring with unity $1_R$.

Examples

For each of the $R$ below, $(R,+_R,\cdot_R )$, where $+_R,\cdot_R=+,\cdot$ unless specified, is a ring:

$R$	Note
$\mathbb{Z}$	Commutative ring with unity. $1_R=1,0_R=0$
$\mathbb{Z}/6\mathbb{Z}=\{0,1,2,3,4,5\}$	Commutative ring with unity.$1_R=1,0_R=0$. In this ring, $a+_Rb=a+b \mod 6; a \cdot_R b=a\cdot b \mod 6$
$\{\begin{pmatrix}a&b\\c&d\end{pmatrix}|a,b,c,d \in \mathbb{R}\}$	Non-commutative ring of real $2\times 2$ matrices with unity. $1_R=\begin{pmatrix}1&0\\0&1\end{pmatrix},0_R=\begin{pmatrix}0&0\\0&0\end{pmatrix}$

Zero divisor

Let $(R,+_R,\cdot_R )$ be a ring.

If elements $a,b \in R, a,b \neq 0_R$ satisfies

$$ a\cdot_R b=0_R $$

then $a,b$ are a pair of zero divisor in ring $(R,+_R,\cdot_R )$.

Example:

In the ring $(\mathbb{Z}/6\mathbb{Z}=\{0,1,2,3,4,5\},a+_Rb=a+b \mod 6; a \cdot_R b=a\cdot b \mod 6$, $2,3$ are a pair of zero divisor as $2 \cdot_R 3= 2\cdot 3 \mod 6 = 0$