I would write $R \sqcup \mathbb{Z}[X]$, since the ring you describe is in fact just this coproduct in the category of rings, or $R * \mathbb{Z}[X]$ (since people do not like the coproduct symbol for some reason when applied to the category of rings and rather speak of about free products). If you want to have something similar to $R[X]$, I would suggest $R\{X\}$ or $R|X|$. I don't know if this is standard notation, probably not. When you use a notation for this ring in a paper or talk, you will probably have to explain it anyway.

I would write $R \sqcup \mathbb{Z}[X]$, since the ring you describe is in fact just this coproduct in the category of rings, or $R * \mathbb{Z}[X]$ (since people do not like the coproduct symbol for some reason when applied to the category of rings and rather speak of about free products). If you want to have something similar to $R[X]$, I would suggest $R\{X\}$ or $R|X|$ or $R \lfloor X \rfloor$. I don't know if this is standard notation, probably not. When you use a notation for this ring in a paper or talk, you will probably have to explain it anyway.

I would write $R \sqcup \mathbb{Z}[X]$, since this is in fact this coproduct, or (since people do not like the coproduct symbol for some reason when applied to the category of rings) $R * \mathbb{Z}[X]$. If you want to have something similar to $R[X]$, I would suggest $R\{X\}$.

I would write $R \sqcup \mathbb{Z}[X]$, since the ring you describe is in fact just this coproduct in the category of rings, or $R * \mathbb{Z}[X]$ (since people do not like the coproduct symbol for some reason when applied to the category of rings and rather speak of about free products). If you want to have something similar to $R[X]$, I would suggest $R\{X\}$ or $R|X|$. I don't know if this is standard notation, probably not. When you use a notation for this ring in a paper or talk, you will probably have to explain it anyway.