A Monoid is a set S with a binary operation '+', with the following properties (for A,B,C in S):
 Closure: A + B is in S
 Associative: A + ( B + C ) = ( A + B ) + C
 Identity: There's an I s.t. A+I = A = I+A

The set of square matrices with multiplication is a Monoid because not every square matrix has an inverse 
A Group is a Monoid, with the following properties:
 Inverse: There's an A⁻¹ s.t. A+A⁻¹ = I = A⁻¹+A

The set of invertible matrices with multiplication is a Group (The General Linear Group) 
A Commutuative Group ( Abelian Group ) is a Group G, with the following properties (for A,B in G):
 Commutative: A + B = B + A

The set of invertible diagonal matrices with multiplication is a Commutative Group because diagonal matrices commute 
A Ring R is a Commutuative Group under '+' and Monoid under '*', with the following properties:
 Distributive: '*' distributes over '+'

The set of square matrices with '*' and '+' is a Ring because matrix multiplication distributes over addition 
A Commutative Ring is Ring, with the following properties:
 Commutative: A * B = B * A

A set of similar matrices with '*' and '+' is a Commutative Ring because similar matrices commute 
A Field is a Commutative Ring, with the following properties:
 Inverse: For A≠0, There's an A⁻¹ s.t. A*A⁻¹ = I = A⁻¹*A

The set of diagonal matrices with '*' and '+' is a Field because nonzero diagonal matrices are invertible 