Isomorphism Of Cross Quotient rings

Question

I have two questions that I need help about.

1. Let $R$ and $S$ be rings, $I$ and $J$ ideals of $R$ and $S$ respectively. Prove that $I\times J$ is an ideal of $R\times S$ and $(R\times S)/(I\times J)$ is isomorphic to $R/I \times S/J$.
2. What is $R/I$ in the following situation: $R = C[0,1]$, $I = \{ f\in R \mid f(1/2)=0 \}$

Answer 1

1. Hint: To see that $I\times J$ is an ideal of $R\times S$, just check the properties of ideals, remembering that every element of $R\times S$ is of the form $(r,s)$ with $r\in R$ and $s\in S$, and that addition and multiplication in $R\times S$ are computed componentwise. As for the quotient isomorphism, consider the [surjective] map \begin{align*} f : R\times S&\to R/I\times S/J\\ (r,s)&\mapsto (r + I, s + J). \end{align*} Examine the kernel and apply the first isomorphism theorem.
2. Hint: I assume you mean $R = C[0,1] = \{f : [0,1]\to\Bbb R\mid f\textrm{ continuous}\}$ and $I = \{f\in R\mid f(1/2) = 0\}$ (you use $R$ twice in what looks like different ways without making a distinction). Consider the [surjective] map \begin{align*} \phi : R&\to\Bbb R\\ f&\mapsto f(1/2). \end{align*} Again, examine the kernel and apply the first isomorphism theorem.

Takeaway: the first isomorphism theorem is a very useful tool in understanding quotient rings.