Looking for a terminology in ring theory ("ideal" which is not necessarily closed under addition )

Question

I am wondering if there is a name for the subsets $S$ of a commutative ring $R$ such that for every $r\in R$ and every $s\in S$ we have $rs\in S$.

Thus $S$ is almost an ideal except that it is not closed under addition.

Example: unions of some ideals in a commutative ring.

Answer 1

In the semigroup theory S is called a left ideal in the semigroup R under multiplication. See any semigroup theory book (J.M.Howie: Semigroup theory for example)