Question about the radical of the Jacobson radical. I am confused about the notation $\operatorname{rad}^2 A$. It can be considered as $\operatorname{rad}(\operatorname{rad}(A))$ or as $(\operatorname{rad}(A))^2$. Are $\operatorname{rad}(\operatorname{rad}(A))$ and $(\operatorname{rad}(A))^2$ the same?
Here $\operatorname{rad}(\operatorname{rad}(A))$ is the radical of the Jacobson radical of $A$ and $(\operatorname{rad}(A))^2$ consists of all finite sums of the elements of the form $x_1 x_2$, where $x_1, x_2 \in \operatorname{rad}(A)$.
Since $(\operatorname{rad}(A))^2$ is a nilpotent ideal of $\operatorname{rad}(A)$ and $\operatorname{rad}(\operatorname{rad}(A))$ is the maximal ideal of $\operatorname{rad}(A)$, we know that $(\operatorname{rad}(A))^2 \subseteq \operatorname{rad}(\operatorname{rad}(A))$. But how to prove that $\operatorname{rad}(\operatorname{rad}(A)) \subseteq (\operatorname{rad}(A))^2$? I have another question: is $\operatorname{rad}(A)$ the unique maximal ideal of $A$? Thank you very much. 
Edit: this question comes from the question. 
 A: I guess you are pretty much confused by the definition.  Assume throughout $A$ is finite dimensional (I suppose it works for artinian rings in general).
The (Jacobson) radical $J(A)$ of an algebra $A$ has many equivalent definitions, which you can simply look up from, for example, wikipedia.  For the radical rad$M$ of a (left) $A$-module $M$, you can use either (1) $\mathrm{rad}M=J(A)M$ or (2) $\mathrm{rad}M$=intersection of maximal submodule of $M$.  Again (1) and (2) are equivalent, by exactly the same reason(s) as why different definitions for radical of $A$ are the same.  If you want more detailed explanation, please consult Auslander-Reiten-Smalo's book "Representation theory of Artin algebras" Proposition 3.5.
So when you consider the radical of the radical of $A$, you are considering the radical of an $A$-module, namely, $M=\mathrm{rad}A$, which if you go by definition (1), will become $\mathrm{rad}A\mathrm{rad}A$.
For your second question, it is false in general, unless $A$ is local.
P.S.  When I first started learning representation theory of algebras, I used both Assem-Skowronski-Simson and ARS at the same time, to complement each other contents; ARS arguments are usually the one that is preferred because it is very concise and the proofs are elegant; ASS has a very down-to-earth, example-rich approach which is very good to understand what is going on concretely, which naturally becomes a very good supplement to ARS. (and it has more interesting theories such as tilting modules and torsion theory)
EDIT: Sorry for being lazy on tex-typing; and missing authors' names.  Added them back in (Assem-Skowronski-Simson wrote book that OP is constantly raising questions from; so I assumed he knows but I forgot about the general audience.)
A: If $B$ is an ideal of $A$ then $\operatorname{rad}(B)=\operatorname{rad}(A)\cap B$ [Herstein, Noncommutative Rings, Theorem 1.2.5]. So 
$\operatorname{rad}(\operatorname{rad}(A))=\operatorname{rad}(A)$.
