A group with an extra operation called multiplication which satisfies:

- associative property
- distributes over addition (the default group operation)
- has an identity

Unlike addition, that multiplication does not need to satisfy:If those are also satisfied, then we have a field.

- commutative property. If this is satisfied, we can call it a commutative ring.
- existence of an inverse. If this is satisfied, we can call it a division ring.

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.

A polynomial ring is another example with the same properties as the integers.

The simplest non-commutative ring that is not a field is the set of all 2x2 matrices of real numbers:Note that $GL(n)$ 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