I was reading a bit about universal algebra recently and was wondering whether there's a "general way" to encode concepts like ring ideals and normal subgroups.
Ring ideals and normal subgroups both have closure properties that refer back to the larger structure that they are part of and I'm wondering if there's a well-known way to refer to properties like this in a universal algebra setting.
My question is similar to this one, but that question specifically asks about the roles that ideals and normal subgroups play in congruences. I am mostly interested in their closure properties and the machinery needed to define them in a general setting.
Is there a semi-standard general technique for talking about subuniverses in "extended theories" that are partially particular to individual structures or a similar notion that achieves the same goals?
Corrections:
As Noah Schweber points out, for this construction to work, we need the constant $1$ to not be part of the ring signature. For the purposes of this question, I'm considering rings to be commutative, but not necessarily have a multiplicative identity.
An ideal $I$ in a ring $R$ is an ideal if and only if it satisfies the following two properties:
$$ I + I = I \;\; \text{and} \;\; RI = I $$
I.e. a prospective ideal $I$ needs to be closed under addition by the elements of $I$ and under multiplication by any ring element.
Similarly, one of the many ways of describing a normal subgroup $N$ of a group $G$ is
$$ N \;\text{is a group} \;\; \text{and} \;\; \text{$N$ is fixed by conjugation by elements of $G$} $$
If we take a ring $R = (R_0, F)$ and add unary functions to $F$ corresponding to multiplication by all elements of $R$, we can get a new theory $R'$. I think $R'$ is technically an algebra (with the caveat that its scalars form a ring and do not in general form a field).
Anyway if we take our expanded structure $R'$, shown below, then I think $I$ corresponds to a subuniverse.
$$ R' = (R_0, F \cup \{ \text{multiplication by $r$} : r \in R_0\}) $$
Similarly, if I take my group $G$ and expand it with unary functions for operations representation conjugation by fixed elements, then I get $G'$.
$$ G' = (G_0, F \cup \{ \text{conjugation by $g$} : g \in G_0 \}) $$
In this setting, I think a normal subgroup is just a subuniverse of $G'$.