4
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ Interesting question. I personally haven't seen such a word used (probably because I've never seen this idea discussed.) $\endgroup$ – rschwieb Jul 1 '15 at 9:54
  • 2
    $\begingroup$ Candidate if there is no term: idel? Ideal without addition? $\endgroup$ – rschwieb Jul 1 '15 at 9:56
  • $\begingroup$ @rschwieb interesting suggestion. But before it, I am interested to know if there exists a terminology already used in the literature. $\endgroup$ – user251694 Jul 1 '15 at 10:04
  • 2
    $\begingroup$ @rschwieb The word idele is already used (even though the spelling is different)... $\endgroup$ – Pierre-Guy Plamondon Jul 1 '15 at 10:54
  • 2
    $\begingroup$ @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 :) $\endgroup$ – rschwieb Jul 1 '15 at 13:01
2
$\begingroup$

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)

$\endgroup$
  • $\begingroup$ This can hardly help imo, as a semigroup has only one operation defined on it and not two, as rings require. $\endgroup$ – Timbuc Jul 1 '15 at 10:05
  • 2
    $\begingroup$ @Timbuc Then again, the definition itself ignores addition, so what more could you expect? $\endgroup$ – Matt Samuel Jul 1 '15 at 10:23
  • $\begingroup$ 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. $\endgroup$ – Timbuc Jul 1 '15 at 10:29
  • $\begingroup$ @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. $\endgroup$ – user251694 Jul 1 '15 at 10:37
  • 1
    $\begingroup$ "left semigroup ideal" sounds like a viable ad-hoc term to use, if nothing better appears. $\endgroup$ – rschwieb Jul 1 '15 at 13:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.