# Special Types of Rings

1. Commutative Rings

A ring $R$ is said to be a commutative if the multiplication composition in $R$ is commutative, i.e.
$ab = ba\,\,\,\forall a,b \in R$

2. Rings With Unit Element

A ring $R$ is said to be a ring with unit element if $R$ has a multiplicative identity, i.e. if there exists an element $R$ denoted by $1$, such that
$1 \cdot a = a \cdot 1\,\,\,\forall a \in R$

The ring of all $n \times n$ matrices with element as integers (rational, real or complex numbers) is a ring with unity. The unity matrix
${I_n} = \left[ {\begin{array}{*{20}{c}} 1&0&0& \cdots &0 \\ 0&1&0& \cdots &0 \\ 0&0&1& \cdots &0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0&0&0& \cdots &1 \end{array}} \right]$ is the unity element of the ring.

3. Rings With or Without Zero Divisors

While dealing with an arbitrary ring $R$, we may find elements $a$ and $b$ in $R$, where neither of which is zero and their product may be zero. We call such elements divisors of zero or zero divisors.

Definition:

A ring element $a\left( { \ne 0} \right)$ is called a divisor of zero if there exists an element $b\left( { \ne 0} \right)$ in the ring such that either
$ab = 0$ or $ba = 0$

We also say that a ring $R$ is without zero divisors if the product of no two non-zero elements of the same is zero, i.e. if
$ab = 0 \Rightarrow$ either $a = 0$ or $b = 0$ or both $a = 0$ and $b = 0$