Maybe we can just give a pointer to section in nLab about R-rings:

Absolutely.

]]>We can separate the section “rings over a ring” to a separate entry (like corings have).

Rings over $R\otimes R^{op}$ are very much used in bialgebroid theory, since Takeuchi. (This is why source and target map appear in role of a unit.)

]]>$R$-rings and $R$-corings are rather standard objects in modern algebra. Maybe we can just give a pointer to section in $n$Lab about $R$-rings: https://ncatlab.org/nlab/show/ring#rings_over_a_ring_rings

Orlov explains somewhere that the case when $R$ is not in a center is needed for his (derived algebraic geometry) purposes.

]]>You should define the term “$R$-ring”, this is non-standard terminology, as far as I am aware. I see that Orlov introduces it on his p. 8, but it remains unclear to me why he can’t just consider any $R$-algebras.

]]>A stub for a notion *different* from twisted tensor product of a dg-algebra with a dg-coalgebra, which has a previously existing entry.