[AntonioLao]

A ring is defined in mathematics as a set that is closed under addition and multiplication. It is an Abelian group for addition and it is an associative semigroup for multiplication, in which the distributive properties hold for both addition and multiplication.

A semigroup is an algebraic system closed under an associative binary operation.

An example of a ring is the Boolean ring. It is a nonempty collection of sets with the properties that the unions and the relative complements of every set are also found in the collection.

A special kind of Hadamard matrices can also form a ring. This kind of algebraic ring is also an Abelian (commutative) group under addition with a given identity and additive inverses. Under rules of matrix multiplication, the elements of the ring do not have inverses hence the multiplicative identity does not exist, while each binary multiplication generates positive integer factors indicating the dimensionality of the sets.

The benefit of these Hadamard matrices is that they can be used to quantize spacetime and describe charge attributes by Abelian groups and describe mass attributes by multiplicative semigroups.

[Guille]

What is the difference between a group and a ring?

[AntonioLao]

the elements of a group have their inverses also in the group. The elements of a ring do not have any multiplicative inverses hence a ring is a semigroup only for multiplication. However, a ring does have additive inverses. They are also Abelian group for addition.