Question

Question: Suppose $\mathit{I}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{J}$and are ideals in a ring $\mathit{R}$and let role="math" localid="1657281911044" $\mathit{f}\mathbf{}\mathbf{:}\mathit{R}\underset{}{\mathbf{\to }}\mathbf{}\raisebox{1ex}{$\mathbf{R}$}\!\left/ \!\raisebox{-1ex}{$$}\right.\mathit{I}\mathbf{\times }\mathbf{}\raisebox{1ex}{$\mathbf{R}$}\!\left/ \!\raisebox{-1ex}{$$}\right.\mathit{J}$be the function defined by $\mathit{f}\mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathit{a}\mathbf{+}\mathit{b}\mathbf{,}\mathit{a}\mathbf{+}\mathit{J}\mathbf{)}$

(c) What is the kernel of ?

Short answer

The kernel of is $\mathit{I}\mathbf{\cap }\mathit{J}$

Step-by-step solution

Kernel of a homomorphism. f:

The kernel of a homomorphism $\mathit{f}\mathbf{:}\mathit{R}\underset{}{\mathbf{\to }\mathbf{R}\mathbf{\text{'}}}$is a set of all elements of $\mathit{R}$which is mapped to identity element of $\mathit{R}\mathbf{\text{'}}$that is if localid="1657283618087" $\mathit{e}\mathbf{\text{'}}$be the identity element of $\mathit{R}\mathbf{\text{'}}$

localid="1657283774610" $\mathit{k}\mathit{e}\mathit{r}\mathbf{\left(}\mathit{f}\mathbf{\right)}\mathbf{=}\mathbf{\{}\mathbf{X}\mathbf{\in }\mathbf{R}\mathbf{:}\mathbf{f}\mathbf{(}\mathbf{x}\mathbf{)}\mathbf{=}\mathbf{e}\mathbf{\text{'}}\mathbf{\}}$

Conclusion 

By the definition of kernel of , if $\mathit{a}\mathbf{\in }\mathbf{}\mathit{k}\mathit{e}\mathit{r}\mathit{f}$then we have,

$\mathbf{(}\mathbf{}\mathit{a}\mathbf{+}\mathit{I}\mathbf{,}\mathit{a}\mathbf{+}\mathit{J}\mathbf{)}\mathbf{=}\mathbf{(}\mathit{I}\mathbf{,}\mathit{J}\mathbf{)}$, which implies that $\mathit{a}\mathbf{\in }\mathit{I}\mathbf{\cap }\mathit{J}$

Therefore, it is clear that $\mathit{k}\mathit{e}\mathit{r}\mathbf{\left(}\mathit{f}\mathbf{\right)}\mathbf{=}\mathit{I}\mathbf{\cap }\mathit{J}$