  Let $R$ be a commutative unital ring, $a\in R$ and $t$ a positive integer.
$a^{t}$-reduced $R$-modules and universally $a^{t}$-reduced $R$-modules are
defined and their properties given. Known (resp. new) results about reduced
$R$-modules are retrieved (resp. obtained) by taking $t=1$ and results about
reduced rings are deduced.
