Quotient Ring
Dependencies:
Let $I$ be an ideal of $R$. The quotient ring $R/I$ is the set of all additive cosets of $I$.
\[ R/I = \{r + I: r \in R\} \]
Here addition is defined as $(a+I) + (b+I) = (a+b) + I$ and multiplication is defined as $(a+I)(b+I) = ab+I$.
Proof that $R/I$ is a ring
$(R/I, +)$ is a factor group. Since $R$ is abelian, $(R/I, +)$ is an abelian group.
Multiplication in $R/I$ is well-defined and its definition implies closure.
\begin{align} & ((a+I)(b+I))(c+I) \\ &= (ab+I)(c+I) \\ &= ((ab)c + I) \\ &= (a(bc) + I) \\ &= (a+I)(bc+I) \\ &= (a+I)((b+I)(c+I)) \end{align}
Therefore, multiplication in $R/I$ is associative.
\begin{align} & (a+I)((b+I)+(c+I)) \\ &= (a+I)((b+c)+I) \\ &= (a(b+c) + I) \\ &= ((ab+ac) + I) \tag{distributivity in $R$} \\ &= (ab+I) + (ac+I) \\ &= (a+I)(b+I) + (a+I)(c+I) \end{align}
Therefore, distributivity holds for $R/I$.
Dependency for:
- I is a prime ideal iff R/I is an integral domain
- I is a maximal ideal iff R/I is a field
- F[x]/p(x) is isomorphic to F[x]/p(x)F[x]
Info:
- Depth: 5
- Number of transitive dependencies: 12