Being even and being a square in arbitrary rings $R$ are analogous by definition:
$p \in R$ is even if there is an $a \in R$ with $p = a + a =: 2a$.
$p \in R$ is a square if there is an $b \in R$ with $p = b \cdot b =: b^2$.
I wonder if there other analogies between the two concepts (than this one by definition).
And which other relations are there between them next to:
(1) When $p$ is a square then $b$ is even iff $p$ is even.
[Note that no analogy arises from "When $p$ is even then $a$ is a square iff $p$ is a square" because the latter is false.]
Furthermore:
How do you prove that when $p$ is an even square, i.e. $p= a + a = b\cdot b$, then $b$ is even, i.e. $b = c + c$?
(The other direction is easy.)