What are perfect squares in general rings? The definition of a square element can be made for arbitrary rings $R$:

$p\in R$ is a square if there is an $a\in R$ with $p = aa$.

The definition of a perfect square element for arbitary rings – inspired by perfect square numbers – would presuppose the concept of an integral element in arbitrary rings:

$p\in R$ is a perfect square if there is an integral element $a\in R$ with $p = aa$.

Note, that for  $R = \mathbb{Z}$ being a square and being a perfect square are equivalent, while for $R = \mathbb{Q}$ it's not.
But the definition of integral elements:

$p\in R$ is integral over $A$, a subring of $R$, if there are $n \geq 1$ and $a_{j}\in A$ such that $p^{n}+a_{n-1}p^{n-1}+\cdots
 +a_{1}p+a_{0}=0$

depends on a subring $A \subset R$. So also the definition of perfect squares depends on a subring $A \subset R$.
My questions are:

  
*
  
*Is there an "absolute" (not relative) definiton of being a perfect square (like there are absolute definitons of being a prime element or a square element)? 
  
*Does the concept of being a perfect square (with respect to a subring $A \subset R$) play an important role in ring theory/algebra?
  
*Are there sensible reasons that the name "perfect square" had been chosen (instead of the more self-explanatory name "integral square")?

 A: A perfect square (in normal use) is an element of $\Bbb Z$ which is the square of an element of $\Bbb Z$. That is, it is a square in $\Bbb Z$. Even if you look in some larger ring, say $\Bbb Q$, then the perfect squares are precisely the elements of $\Bbb Z$ which are squares in $\Bbb Z$. The fact that $\Bbb Z$ is a subring of $\Bbb Q$ is crucial to the definition - indeed, the point of the term "perfect square" is that for $\Bbb Z \subset R$ - e.g. $R = \Bbb C, \Bbb R, \Bbb Q$ - then the set of "perfect squares" in $R$ is the same for any choice of $R$ - it's the set of squares in $\Bbb Z$. 
So why we have the term "perfect square" at all? Fundamentally, it's because if you call them "square numbers" then you get people saying "but $2$ is a square number it's the square of $\sqrt 2$". "Integral square" would work, except that this a concept you teach to 8 year olds and "perfect", unlike "integral", is a word they already know.
But if you're at the point where you can say "a square in the ring $R$", then the term "perfect square" isn't adding any mathematical content: it's just shorthand for "a square in the ring $\Bbb Z$".
