Question

Suppose I and J are ideals in a ring R and let $f:R\to \frac{R}{I}\times \frac{R}{J}$be the function defined by$f\left(a\right)=\left(a+I, a+J\right)$

1. Prove that fis a homomorphism of rings.

Short answer

It is proved that is a homomorphism of rings.

Step-by-step solution

Homomorphism of rings

Let R and $R\text{'}$be two rings. A mapping $f:R\to R\text{'}$is called an homomorphism of rings if,

$f\left(a+b\right)=f\left(a\right)+f\left(b\right)$

$f\left(ab\right)=f\left(a\right)f\left(b\right)$ for all$a,b\in R$

Proof

Since I and J are ideals in a ring R therefore, for every$a,b\in R$

$\begin{array}{rcl}f\left(a+b\right)& =& \left(\left(a+b\right)I, \left(a+b\right)J\right)\\ & =& \left(a+I, a+J\right)+\left(b+I, b+J\right)\\ & =& f\left(a\right)+f\left(b\right)\\ & & \end{array}$

And,

$\begin{array}{rcl}f\left(ab\right)& =& \left(ab+I, ab+J\right)\\ & =& \left(a+I, a+J\right)\left(b+I, b+J\right)\\ & =& f\left(a\right)f\left(b\right)\\ & & \end{array}$

Clearly, f is a homomorphism of rings