Unital ring has part(s) of the class monoid
Definition
A ring is a set R equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms
R is an abelian group under addition, meaning that:
(a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative).
a + b = b + a for all a, b in R (that is, + is commutative).
There is an element 0 in R such that a + 0 = a for all a in R (that is, 0 is the additive identity).
For each a in R there exists −a in R such that a + (−a) = 0 (that is, −a is the additive inverse of a).
R is a monoid under multiplication, meaning that:
(a · b) · c = a · (b · c) for all a, b, c in R (that is, ⋅ is associative).
There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R (that is, 1 is the multiplicative identity). 
Multiplication is distributive with respect to addition, meaning that:
a · (b + c) = (a · b) + (a · c) for all a, b, c in R (left distributivity).
(b + c) · a = (b · a) + (c · a) for all a, b, c in R (right distributivity).Category-theoretic description
Every ring can be thought of as a monoid in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product of 
  
    
      Z
    
    \mathbb {Z}
  -modules). The monoid action of a ring R on an abelian group is simply an R-module. Essentially, an R-module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a "vector space over a ring".
Let (A, +) be an abelian group and let End(A) be its endomorphism ring (see above). Note that, essentially, End(A) is the set of all morphisms of A, where if f is in End(A), and g is in End(A), the following rules may be used to compute f + g and f ⋅ g: