Unital ring has part(s) of the class abelian group
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).Module
The concept of a module over a ring generalizes the concept of a vector space (over a field) by generalizing from multiplication of vectors with elements of a field (scalar multiplication) to multiplication with elements of a ring. More precisely, given a ring R, an R-module M is an abelian group equipped with an operation R × M → M (associating an element of M to every pair of an element of R and an element of M) that satisfies certain axioms. This operation is commonly denoted by juxtaposition and called multiplication. The axioms of modules are the following: for all a, b in R and all x, y in M,