1. Commutative Rings:
A ring is said to be a commutative, if the multiplication composition in is commutative. i.e.
2. Rings with Unit Element:
A ring is said to be a ring with unit element if has a multiplicative identity, i.e. if there exist an element denoted by , such that
The ring of all 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 , we may find elements and in 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 is called a divisor of zero if there exist an element in the ring such that either
We also say that a ring is without zero divisors if the product of no two non-zero elements of same is zero, i.e. if
either or or both and