Maximal subideal of an ideal For a commutative ring $R$ with unity, I am looking for an equivalent condition for an ideal $T$ to have the property that $T$ contains a unique maximal proper subideal, equivalently, the sum of proper subideals of $T$ is not equal to $T$.
 A: Claim: 
Let $T$ be an ideal in a commutative ring. If $T$ contains
a unique maximal proper subideal, then $T$ is principal. 
Proof: Indeed take any element outside the unique maximal proper subideal. The ideal generated by that cannot be a proper subideal and hence has to be $T$.
So, this suggests that this is a very special property. For instance, that maximal subideal would have to contain the square of any generator of $T$ and if those generators are not zero-divisors, then they are associate (i.e. they can only differ by a unit multiple). If any generator is a zero-divisor then every element in $T$ is a zero-divisor. I think you can continue this line of thought to derive more special properties when this holds. 
There is also a partial converse:
Claim:
Let $T$ be a non-zero principal ideal in a commutative ring. Then every proper subideal of $T$ is contained in an ideal that is maximal among proper subideals in $T$. 
(By the usual proof of existence of maximal ideals)
Proof: Let $T\subseteq R$ be a principal ideal generated by $t\in T$, $T'\subsetneq T$  a proper subideal, and consider the set of ideals $I\subseteq T$ such that $T'\subseteq I$, $t\not\in I$ which is the same as the set of proper subideals of $T$ containing $T'$. If there is a chain of such ideals $\ldots\subseteq I_\lambda\subseteq \dots$ with $\lambda\in \Lambda$, then $t\not\in J:=\cup_{\lambda\in\Lambda}I_\lambda$ is an upper bound of the chain and hence the statement follows by Zorn's lemma.
However, of course, this does not mean that there is a unique one...
On the positive side, here is a situation where what you are hoping for holds:
Claim: If $R$ is a DVR, then the ideals in $R$ form a totally ordered set.
Proof: This follows essentially from the definition of a DVR. 
So, I suppose one can conclude that this holds sometime, but for most ideals in most rings it does not. 
