Let $R$ be a noncommutative ring (with unit). Then a "fully noncommutative" (for a lack of better wording) monomial over $R$ in the single noncommutative indeterminate $X$ of degree $d$ is given by a finite sum of words of the form $$ r_0 X r_1 X...X r_{d-1} X r_d,$$ where $r_0,...,r_d\in R$. Then one constructs the ring of "fully noncommutative" polynomials in $X$ in the obvious way. Analogously, one gets "fully noncommutative" polynomials and formal power series over $R$ in several or infinitely many noncommutative variables. (Note that this construction should not be confused with $R\langle X,Y\rangle$, where the variables do not commute with each other but are assumed to commute with the $R$-coefficients.)

(Q1) Is there a standard notation for these rings?

Currently I'm using the notation $R\rangle X\langle$, resp. $R\rangle X,Y\langle$, as I need to distinguish between the latter and $R[S,T]$ and $R\langle X,Y\rangle$. But it looks... odd and I have no desire to introduce possibly non-standard notation. Another possibility is to use $\langle R,X\rangle$ (since the elements are words), but it is not particularly suggestive of what's supposed to be a ring of polynomials and reminds more of other things (it's unfortunately overused notation).

Moreover, I imagine such rings have been well studied in the literature, but I was unable to locate them (one usually finds only the $R\langle X,Y\rangle$-type).

(Q2) What are some standard references on "$R\rangle X\langle$" ?

Thanks for your time!