Image-preimage adjunction induced from regular epi respects regular monos?

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Concretely, every set function $f:A\rightarrow B$ induces an adjunction $f_\ast \dashv f^\ast$ between image and preimage. For rings groups, the image and preimage along a surjective ring group homomorphism preserve ideals normal subgroups, which are equalizers of arrows with the arrow that picks out $0$. This is no surprise for $f^\ast$ since it preserves equalizers, but not generally true for $f_\ast$.

Since rings groups are an algebraic theory, the regular epis are precisely the surjective functions. It seems one could just say that regular epis induce image-preimage adjunctions that preserve regular monos.

As pointed out in the comments, the case of rings is not as pleasant because ideals leave the category of rings, so maybe I should replace 'regular' by 'normal'...

At any rate, in which algebraic categories (with zero object) does this hold?

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Firstly the comments are perhaps misplaced regarding rings since it depends whether or not the definition you use requires a multiplicative identity or not.

For normal epimorphisms and normal monomorphisms this condition has been studied and such categories are called Ideal-determined categories. Any variety of universal algebras that has a unique constant $e$ and for some natural number $n$ one $n+1$-ary term $p(x_1,..,x_n,x_{n+1})$ and $n$ binary terms $s_i(x,y)$ such that $s_i(x,x)=e$ and $p(s_1(x,y),..,s_n(x,y),y)=x$ satisfies this condition. These include the categories of groups, Lie algebras over a field (or even a commuative ring), not necessarily unital rings and many others. It is perhaps worth noting that there are examples that satisfy this condition that do not have such terms.