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 exists 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 , 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 is called a divisor of zero if there exists 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 the same is zero, i.e. if
either or or both and