Show that a ring can't be expressed as union of 2 proper ideals but it is possible to express it as a union of three proper ideals.
My solution
First Part
Let$ R$ be the ring with $ A$ , $B$ & $C$ as proper ideals
Assuming $ R=A\cup_{}B$
As $R$ is an ideal of itself -> $A\cup_{}B$ is also an Ideal which is possible only if
(i) $A$ is contained in $B$
->$ B = R$ -> $B$ is an improper ideal -> contradiction
or
(ii) $B$ is contained in $A$ .
-> $A = R$ -> $A$ is an improper ideal -> contradiction Proved.
Second part
Let $ R=A\cup_{}B\cup_{}C $
(i) $C$ is contained in $A\cup_{}B$
-> $A\cup_{}B = R$ . This is similar to the above part which results in contradiction
(ii) $A\cup_{}B$ is contained in $C$ .
-> $C = R$ -> $C$ is an improper ideal -> contradiction
I am not able to solve the second part. What is wrong with my approach?