  The primary purpose of the present paper is to investigate when relations of
the types $|AB|=|A||B|$, $|A\pm B|\leq |A|+|B|$, $||A|-|B||\leq |A\pm B|$ and
$|\overline{\text{Re} A}|\leq |A|$ (among others) hold in an unbounded operator
setting. As interesting consequences, we obtain a characterization of
(unbounded) self-adjointness as well as a characterization of invertibility for
the class of unbounded normal operators.
