Question

Let $\mathit{L}$and $\mathit{K}$be ideals in a ring $\mathit{R}$, with $\mathit{K}\underline{\mathbf{\subset }}\mathit{I}$. Prove that $\mathit{I}\mathbf{/}\mathit{K}\mathbf{=}\mathbf{\{}\mathit{a}\mathbf{+}\mathit{K}\mathbf{:}\mathit{a}\mathbf{\in }\mathit{I}\mathbf{\}}$is an ideal in the quotient ring. $\mathit{R}\mathbf{/}\mathit{K}$

Short answer

It is proved that. $\mathit{I}\mathbf{/}\mathit{K}\mathbf{=}\mathbf{\{}\mathit{a}\mathbf{+}\mathit{K}\mathbf{:}\mathit{a}\mathbf{\in }\mathit{I}\mathbf{\}}$is an ideal in the quotient ring$\mathit{R}\mathbf{/}\mathit{K}$

Step-by-step solution

Theorem

A non-empty subset $\mathit{I}$localid="1657379416911" $\mathit{K}$of a ring$\mathit{R}$is an ideal if and only if it has these properties:

1. If $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{I}$then, $\mathit{a}\mathbf{-}\mathit{b}\mathbf{\in }\mathit{I}$
2. If $\mathit{r}\mathbf{\in }\mathit{R}$and $\mathit{a}\mathbf{\in }\mathit{I}$then, and $\mathit{a}\mathit{r}\mathbf{\in }\mathit{I}$

Quotient Ring

Let $\mathit{R}$be a ring and $\mathit{I}$be its ideal Then, the set of all cosets of forms an additive commutative group in which addition is defined by

localid="1657295285405" $\mathbf{(}\mathit{a}\mathbf{+}\mathit{b}\mathbf{)}\mathbf{+}\mathbf{(}\mathit{b}\mathbf{+}\mathit{I}\mathbf{)}\mathbf{=}\mathbf{(}\mathit{a}\mathbf{+}\mathit{b}\mathbf{)}\mathbf{+}\mathit{I}$,for $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{R}$and multiplication is defined by

$\mathbf{(}\mathit{a}\mathbf{+}\mathit{I}\mathbf{)}\mathbf{+}\mathbf{(}\mathit{b}\mathbf{+}\mathit{I}\mathbf{)}\mathbf{=}\mathit{a}\mathit{b}\mathbf{+}\mathit{I}$

Proof

Given that,

Firstly, we know that, $\mathbf{0}\mathbf{\in }\mathit{I}\mathbf{/}\mathit{K}\mathbf{,}$which shows that is non-empty.

Let, $\mathit{a}\mathbf{+}\mathit{K}\mathbf{,}\mathit{b}\mathbf{+}\mathit{K}\mathbf{\in }\mathit{I}\mathbf{/}\mathit{K}$be arbitrary then, we get.

$\mathbf{(}\mathit{a}\mathbf{+}\mathit{K}\mathbf{)}\mathbf{-}\mathbf{(}\mathit{b}\mathbf{+}\mathit{K}\mathbf{)}\mathbf{=}\mathbf{(}\mathit{a}\mathbf{-}\mathit{b}\mathbf{)}\mathbf{+}\mathit{K}\mathbf{\in }\mathit{I}\mathbf{/}\mathit{K}$

Now, assume there is some for which,

$\mathbf{(}\mathit{a}\mathbf{+}\mathit{K}\mathbf{)}\mathbf{(}\mathit{r}\mathbf{+}\mathit{K}\mathbf{)}\mathbf{=}\mathit{a}\mathit{r}\mathbf{+}\mathit{K}\mathbf{\in }\mathbf{}\mathit{I}\mathbf{/}\mathit{K}$and

$\mathbf{(}\mathit{r}\mathbf{+}\mathit{K}\mathbf{)}\mathbf{(}\mathit{a}\mathbf{+}\mathit{K}\mathbf{)}\mathbf{=}\mathit{r}\mathit{a}\mathbf{+}\mathit{K}\mathbf{\in }\mathbf{}\mathit{I}\mathbf{/}\mathit{K}$.