# 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.

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 exist an element $R$ denoted by $1$, such that

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

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 $R$ 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 exist 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 same is zero, i.e. if
$ab = 0 \Rightarrow$ either $a = 0$ or $b = 0$ or both $a = 0$ and $b = 0$