Sure, I can provide you with definitions for a Ring and a Group and give you a sample for each.

A Group is a set of elements together with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied: closure, associativity, identity and invertibility. The binary operation, denoted here by $*$, satisfies the following conditions:
1. Closure: For all $a, b$ in the group, $a*b$ is also in the group. 
2. Associativity: For all $a, b$ and $c$ in the group, $(a*(b*c)$. 
3. Identity: There exists an element $e$ in the group such that for all $a$ in the group, $a*e = e*a = a$. 
4. Invertibility: For every element $a$ in the group, there exists an element $b$ in the group such that $a*b = b*a = e$.

One example of a group is the set of integers under addition $(\mathbb{Z},+)$, which satisfies all the group axioms.

A Ring is a set of elements together with two binary operations, addition and multiplication, that satisfy several conditions called ring axioms. A ring is a group under addition, and the multiplication operation satisfies two additional conditions:
1. Distributivity: For all $a, b,$ and $c$ in the ring, $a\cdot (b+c) = a\cdot b + a\cdot c$ and $(a+b)\cdot c = a\cdot c + b\cdot c$.
2. Associativity: For all $a, b,$ and $c$ in the ring, $(a\cdot b)\cdot c = a\cdot (b\cdot c)$.

One example of a ring is the set of integers under addition and multiplication $(\mathbb{Z},+,\cdot)$.

I hope this helps clarify the difference between a Ring and a Group and provides you with some examples to better understand the concepts.
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