Question

Let ${\mathit{l}}_{\mathbf{1}}\mathbf{,}{\mathit{l}}_{\mathbf{2}}\mathbf{,}{\mathit{l}}_{\mathbf{3}}$be ideals in a ring R with identity such that localid="1660489663842" ${\mathit{l}}_{\mathbf{i}}\mathbf{+}{\mathit{l}}_{\mathit{j}}\mathbf{=}\mathit{R}$whenever $\mathit{i}\mathbf{\ne }\mathit{j}$. If ${\mathit{a}}_{\mathbf{i}}\mathbf{\in }\mathit{R}$, prove that the system

$$\begin{gathered} \mathit{x}\mathbf{=}{\mathit{a}}_{\mathbf{1}}\mathbf{(}\mathit{m}\mathit{o}\mathit{d}\mathbf{}{\mathit{l}}_{\mathbf{1}}\mathbf{)} \\ \mathit{x}\mathbf{=}{\mathit{a}}_{\mathbf{2}}\mathbf{(}\mathit{m}\mathit{o}\mathit{d}\mathbf{}{\mathit{l}}_{\mathbf{2}}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.} \\ \mathit{x}\mathbf{=}{\mathit{a}}_{\mathbf{r}}\mathbf{(}\mathit{m}\mathit{o}\mathit{d}\mathbf{}{\mathit{l}}_{\mathbf{r}}\mathbf{)}\mathbf{} \end{gathered}$$

has a solution and that any two solutions are congruent modulo ${\mathit{l}}_{\mathbf{1}}\mathbf{\cap }{\mathit{l}}_{\mathbf{2}}\mathbf{\cap }{\mathit{l}}_{\mathbf{3}}$, [Hint: If s is a solution of the first two congruences, use Exercise 5 and Theorem 14.3 to show that the system

localid="1659198767299" $$\begin{gathered} \mathit{x}\mathbf{=}\mathit{s}\mathbf{(}\mathit{m}\mathit{o}\mathit{d}\mathbf{}{\mathit{l}}_{\mathbf{1}}\mathbf{\cap }{\mathit{l}}_{\mathbf{2}}\mathbf{}\mathbf{)} \\ \mathit{x}\mathbf{=}{\mathit{a}}_{\mathbf{3}}\mathbf{}\mathbf{(}\mathit{m}\mathit{o}\mathit{d}\mathbf{}{\mathit{l}}_{\mathbf{3}}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{} \end{gathered}$$

has a solution, and it is a solution of the original system.]

Short answer

The required system is proved.

Step-by-step solution

Describe the Chinese Remainder Theorem

Consider the relatively prime positive integers ${\mathit{m}}_{\mathbf{1}}\mathbf{,}{\mathit{m}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathit{m}}_{\mathbf{r}}$and the integers $a_{\mathbf{1}}\mathbf{,}{\mathit{a}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathit{a}}_{\mathbf{r}}$and the following system of the congruence equations.

$$\begin{gathered} \mathit{x}\mathbf{=}{\mathit{a}}_{\mathbf{1}}\mathbf{(}\mathit{m}\mathit{o}\mathit{d}\mathbf{}{\mathit{l}}_{\mathbf{1}}\mathbf{)} \\ \mathit{x}\mathbf{=}{\mathit{a}}_{\mathbf{2}}\mathbf{(}\mathit{m}\mathit{o}\mathit{d}\mathbf{}{\mathit{l}}_{\mathbf{2}}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.} \\ \mathit{x}\mathbf{=}{\mathit{a}}_{\mathbf{r}}\mathbf{(}\mathit{m}\mathit{o}\mathit{d}\mathbf{}{\mathit{l}}_{\mathbf{r}}\mathbf{)}\mathbf{} \end{gathered}$$

This system has a solution. Also, if one of the solutions is t, then an integer z is also a solution of this system if and only if the congruence equation $\mathit{z}\mathbf{=}\mathit{t}\mathbf{}\mathbf{(}{\mathit{m}}_{\mathbf{1}}\mathbf{,}{\mathit{m}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathit{m}}_{\mathbf{r}}\mathbf{)}$holds true.

Prove the given system

Consider the following system of congruence equations.

$$\begin{gathered} x=a_1(mod{l}_1) \\ x=a_2(mod{l}_2) \\ x=a_{\mathrm{r}}(mod{l}_{\mathrm{r}}) \end{gathered}$$

Apply the Chinese Remainder Theorem on the first two equations of the given system.

This implies that those two equations have the following solution.

$x=a_4(mod{l}_1\cap l_2)$

Any two solutions of the first two congruence equations are congruent modulo $l_1\cap l_2$.

Apply the Chinese Remainder Theorem on the third equations of the given system.

This implies that those two equations have the following solution.

$x=a_5(mod{l}_1\cap l_2\cap l_3)$

The system of the three equations has a solution and any two solutions are congruent modulo $l_1\cap l_2\cap l_3$.

Therefore, the required result is proved.