Quotient Ring

Definition

Given a Ring $(R,+,\cdot )$ and an Ideal $I$, we can define operations on $R/I$ that make it a ring, called the quotient ring.

Definition $R/I$

• Addition: The addition of two equivalence classes* $[a]$ and $[b]$ in $R/I$ is defined as $[a]+[b]=[a+b]$.
• Multiplication: The multiplication of two equivalence classes* $[a]$ and $[b]$ in $R/I$ is defined as $[a]\cdot [b]=[a\cdot b]$. *For more on equivalence classes, see Equivalence Relation.

The quotient ring simplifies complex algebraic structures into a more manageable form and is useful in deriving important concepts such as Ring Isomorphism.