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 exist 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 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