In Abstract Algebra, rings are algebraic structures that generalize the arithmetic of integers. A Ring ($R$) is a set equipped with two binary operations: addition and multiplication. A fundamental concept in ring theory is the Ideal, which plays a role analogous to normal subgroups in group theory.

An Ideal ($I$) is a subset of a ring $R$ that forms an additive subgroup and has a special “absorbing” property: for any element $r \in R$ and any element $x \in I$, the product $rx$ is always inside $I$.

Quotient Rings are constructed by “modding out” by an ideal. The elements of the quotient ring $R/I$ are the Cosets of the ideal, denoted as $a + I = \{ a + x \mid x \in I \}$. Even though these elements are sets, we can define addition and multiplication on them:

• Addition: $(a + I) + (b + I) = (a + b) + I$
• Multiplication: $(a + I) \cdot (b + I) = (a \cdot b) + I$

In the specific case of the ring of integers modulo $n$ ($Z_n$), every ideal is principal, meaning it is generated by a single element $k$. The ideal generated by $k$, denoted $\langle k \rangle$, consists of all multiples of $k$ modulo $n$. The quotient ring $Z_n / \langle k \rangle$ is isomorphic to $Z_{\gcd(n,k)}$.