Rongzhou: Created page with "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..."

2026-10-17T12:02:02Z

Created page with "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..."

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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 operation|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'''.

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

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

== 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:

{| class="wikitable"
|-
! $R$
! 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 $2 \cdot_R 3= 2\cdot 3 \mod 6 = 0$

Ring (mathematical concept) - Revision history