  Let $A,B\in B(H)$. In the present paper, we establish simple and interesting
facts on when we have $|A||B|=|B||A|$, $|AB|=|A||B|$, $|A\pm B|\leq |A|+|B|$,
$||A|-|B||\leq |A\pm B|$ and $\||A|-|B|\|\leq \|A\pm B\|$, where $|\cdot|$
denotes the absolute value (or modulus) of an operator. The results give some
other interesting consequences.
