  Let $R$ be a noncommutative ring, and let $S$ be an $m$-system of $R$. In
this paper, we give more results on the concept of almost prime (right) ideals,
that were introduced by the first two authors, especially in (right) $S$-unital
rings, local rings, and decomposable rings. In addition, we introduce the
concept of almost right $S$-prime ideals, and we show how some findings
regarding almost prime ideals can be derived as consequences of almost right
$S$-prime ideals. Besides, we show how almost right $S$-prime ideals behave in
related rings such as homomorphic images, quotient rings, and decomposable
rings. Finally, we construct almost right $S$-prime ideals using the Nagata
method of idealization.
