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