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.


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 $$b = 0$$