Let $R := \mathbb{Z}[x,y,x^{-1},y^{-1}]$.
Let $I\subset R$ be the ideal $\langle x+1,y-1,5\rangle$
Let $I'\subset R$ be the ideal $\langle x+1,y-1\rangle$.
Let $j : I\hookrightarrow R$ and $j' : I'\hookrightarrow R$ be the natural inclusions.
Are $j\otimes R/(x-1,y-1)$ and $j'\otimes R/(x-1,y-1)$ injective?
To convince you this is not a homework problem, I'm trying to compute a certain Koszul homology group for certain modules under a group algebra, which is linked to a certain Tor group. I've established a connection between the vanishing of this group and certain properties of some finite groups I'm studying. Properties of these finite groups suggest that at least $j\otimes R/(x-1,y-1)$ should be injective, but I can't convince myself that we must have $(y-1)\otimes 1 = 0$ in $I\otimes_R R/(x-1,y-1)$ (it certainly maps to 0 in $R/(x-1,y-1)$.
My theory doesn't predict any answer for whether $j'\otimes R/(x-1,y-1)$ should be injective.