Do non-additive subgroup ideals of rings exist?

Question

Let $(R,+,\cdot)$ be a ring. A subset $I$ of $R$ is called an ideal if it satisfies

• $(I,+)$ is a subgroup of $(R,+)$
• $I$ "absorbs" products in $R$, that is, for every $x\in I$ and every $a\in R$, both $ax$ and $xa$ are in $I$

My question is: what would we call a subset $I$ that does not satisfy the "additive subgroup" condition?

I can´t find an answer anywhere. I think these type of subsets should also have some interesting properties? Does there exist any theorem about them?

Answer 1

I think you are describing a (monoid) ideal.