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

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.

• Interesting question. I personally haven't seen such a word used (probably because I've never seen this idea discussed.) – rschwieb Jul 1 '15 at 9:54
• Candidate if there is no term: idel? Ideal without addition? – rschwieb Jul 1 '15 at 9:56
• @rschwieb interesting suggestion. But before it, I am interested to know if there exists a terminology already used in the literature. – user251694 Jul 1 '15 at 10:04
• @rschwieb The word idele is already used (even though the spelling is different)... – Pierre-Guy Plamondon Jul 1 '15 at 10:54
• @Pierre-GuyPlamondon I remembered "adele" but I must have forgotten "idele" :) Yes, "idel" would be altogether too close to those. The temptation to mimic "rng" and "rig" was too great for me :) – rschwieb Jul 1 '15 at 13:01

## 1 Answer

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)

• This can hardly help imo, as a semigroup has only one operation defined on it and not two, as rings require. – Timbuc Jul 1 '15 at 10:05
• @Timbuc Then again, the definition itself ignores addition, so what more could you expect? – Matt Samuel Jul 1 '15 at 10:23
• Oh, I expect nothing. I only think this answer might not be what the OP is looking for as it does not use the fact we're in a ring', an algebraic structure with two binary operations defined on it. That's all. – Timbuc Jul 1 '15 at 10:29
• @MattSamuel In fact I am interested in a terminology used for such objects in ring theory. Note that as $S$ lives in a ring it is necessarily a union of ideals. – user251694 Jul 1 '15 at 10:37
• "left semigroup ideal" sounds like a viable ad-hoc term to use, if nothing better appears. – rschwieb Jul 1 '15 at 13:03