Incoming links: Abelian group
A ring where multiplication is commutative and there is always an inverse.
A field can be seen as an Abelian group that has two group operations defined on it: addition and multiplication.
And then, besides each of the two operations obeying the group axioms individually, and they are compatible between themselves according to the distributive property.
Basically the nicest, least restrictive, 2-operation type of algebra.
Examples:
A Ring can be seen as a generalization of a field where:
- multiplication is not necessarily commutative. If this is satisfied, we can call it a commutative ring.
- multiplication may not have inverse elements. If this is satisfied, we can call it a division ring.
Addition however has to be commutative and have inverses, i.e. it is an Abelian group.
The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.
A polynomial ring is another example with the same properties as the integers.
The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:Note that is not a ring because you can by addition reach the zero matrix.
- we know that 2x2 matrix multiplication is non-commutative in general
- some 2x2 matrices have a multiplicative inverse, but others don't