I was taking a math GRE and found a curious property in a question.
There exists subsets $M$ of Rings $R$ that are absorbing but are not ideals.
A prototypical example (the one from the exam) is if given two ideals $U,V$ the set
$$ U \dot V = \lbrace u*v \ u \in U, v \in V \rbrace$$
So naturally take $m \in UV$ take $d \in R$ then it follows $md \in UV$ but $U\dot V$ isn't closed under addition. What is the name of such structures? And are there any interesting theorems regarding them (say like the isomorphism theorems)? Does there exist a general classification of such structures?
IIUC you are asking about subsets $S\subseteq R$ with the property $$ rs\in S\text{ for all }r\in R,\;s\in S\hspace{10mm}(1) $$ Note that the addition operation on $R$ is irrelevant here. Looking only at the multiplication operation, $R$ is a semigroup. In fact a subset of a semigroup satisfying (1) is also called a (semigroup) ideal. There are results about the structure of semigroups and their ideals; see Green's relations.