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(In the scenario I have in mind, rings need not be unital.)

The following notion has come up in some joint work that is being written up. Let $R$ and $S$ be rings, and let $D$ be a subring of $R$. Is there standard notation or terminology for functions $\theta:R\to S$ that satisfy $$\theta(ar)=\theta(a)\theta(r) \quad,\quad \theta(rb)=\theta(r)\theta(b) \quad\hbox{for all $a,b\in D$ and all $r\in R$}\;? \qquad\qquad(*) $$ My first thought is that this is very similar to being a $D$-module map, so that the set of such functions might plausibly be denoted by ${\rm Hom}_D(R,S)$ with some mild abuse of notation. However, if $R=S$ then this notion would be seriously misleading, since then ${\rm Hom}_D(R,R)$ would naturally be interpreted as the set of $D$-module maps from $R$ to itself, which is not what the definition in $(*)$ says!

Note that I am not really asking for people's opinions on inventing new terminology, but checking among the research community if there is an existing accepted name for these objects, which my coauthors and I should therefore follow.

(I'm adding the OA tag even though there is no topology mentioned above, just because there might be workers in operator algebras who have run into these kinds of functions.)

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    $\begingroup$ I don't know a name for this but notice that your condition can be reformulated as follows : $\theta$ restricts to a ring homomorphism on $D$ and if you give $R$ the obvious $D$-bimodule structure and you give $S$ the one induced by the ring homomorphism $\theta|_D$ then $\theta$ is a $D$-bimodule homomorphism. $\endgroup$
    – Benjamin Steinberg
    Commented Jul 13, 2021 at 13:20
  • $\begingroup$ @BenjaminSteinberg That's a good point, which I think I had observed in a vague way in passing but which I hadn't written down. What is slightly odd/annoying is that the map itself has to define the module structure, so as I said in the question ${\rm Hom}_D$ doesn't feel quite right. And, as I keep trying to explain to undergraduates, you can't really say $\theta \in {\rm Hom}_{\theta,D}$... $\endgroup$
    – Yemon Choi
    Commented Jul 13, 2021 at 14:08
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    $\begingroup$ hom_D also doesn't make clear it is a bimodule map and is a ring homomorphism on D $\endgroup$
    – Benjamin Steinberg
    Commented Jul 13, 2021 at 14:10

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