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Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) akin to the theory of ideals?

Note: One can simply forget the additive structure (resp. vector space structure) and consider $R$ as a semigroup with multiplication. In that case, $B$ is a semigroup ideal of $R$. I wonder whether there is a theory that takes into account the addition (resp. the vector space structure) of $R$, as well?

For example, let's say $B$ generates the ring, resp. $B$ spans the algebra. What more can be said about $B$, beyond what the semigroup theory provides?

Any references would be well appreciated.

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  • $\begingroup$ It seems to me that this is just saying that $B$ is a sub-bi-$R$-module of $R$. EDIT: Oh, you're not assuming that $B$ is closed under addition. In that case, as you say, it seems that the additive structure of $R$ is playing no role, so this is really a question about semigroups. $\endgroup$
    – LSpice
    Commented Nov 11, 2021 at 3:54
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    $\begingroup$ You may want to read about "multiplicative ideal theory". In particular, I would recommend to look at Halter-Koch's book on ideal systems. $\endgroup$
    – Salvo Tringali
    Commented Nov 11, 2021 at 7:23
  • $\begingroup$ Crossposted $\endgroup$
    – rschwieb
    Commented Nov 11, 2021 at 14:20
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    $\begingroup$ Let me point out for a simple ring every nonzero semigroup ideal spans the algebra but it is not usually the case that the converse holds. The converse holds for division rings and purely infinite simple rings. $\endgroup$
    – Benjamin Steinberg
    Commented Nov 12, 2021 at 13:15
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    $\begingroup$ No I meant for a simple ring a being 0-simple as a semigruop is equivalent to being a division ring or purely infinite. $\endgroup$
    – Benjamin Steinberg
    Commented Nov 12, 2021 at 16:43

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