Introduction to Rings
Definition of a Ring
Commutative Rings
A ring is commutative if multiplication is commutative: for all .
Ring with Unity
A ring has unity (identity) if there exists such that for all .
Examples
Example 1: is a commutative ring with unity.
Example 2: is a commutative ring with unity ().
Example 3: (n à n matrices) is a non-commutative ring for .
Example 4: (polynomials) is a commutative ring with unity.
Basic Properties
Zero Divisors
A ring without zero divisors is called an integral domain.
Units
Ring Homomorphisms
Kernel and Image
The kernel is an ideal of .
The image is a subring of .