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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?

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    Not that I can think of. Really the interesting thing about ideals comes from the fact that you can mod out by them. They also make nice examples of modules too, but both of these require it to be an additive subgroup. – 2017-02-24
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    Fix $n\in\mathbb{N}$, and set $I=\{0\}\cup\{i\in\mathbb{Z}:|i|\ge n\}$. This is a subset of $\mathbb{Z}$ satisfying the second condition. – 2017-02-24
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    The union of any family of ideals is an "absorbing subset", but not an ideal in general. – 2017-02-24
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    @Adren but are they a "thing" by themselves (not using other subsets in the definiton)? – 2017-02-24
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    @Pedro: I am not aware of any special terminology for "absorbing subsets" of a ring. I guess that this sole property is not enough to produce "interesting" objects. – 2017-02-24
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    @Adren well i hope I find some :). Thank you anyways – 2017-02-24

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I think you are describing a (monoid) ideal.