Let $R$ be a (noncommutative) unital ring with no zero-divisors let and $\mathcal{N}$ a non-zero right module that is projective and torsion-free. Projectivity of course implies that $\mathcal{N}$ is flat, but does projectivity together with torsion-freeness suffice to imply that $\mathcal{N}$ is faithfully flat?

Let $R$ be a (noncommutative) unital ring which is a domain and let $\mathcal{N}$ be a non-zero projective (right) module. Projectivity of course implies that $\mathcal{N}$ is flat, but does the fact that $R$ is also a domain imply that $\mathcal{N}$ is faithfully flat?

Let $R$ be a (noncommutative) unital ring and $\mathcal{N}$ a non-zero right module that is projective and torsion-free. Projectivity of course implies that $\mathcal{N}$ is flat, but does projectivity together with torsion-freeness suffice to imply that $\mathcal{N}$ is faithfully flat?

Let $R$ be a (noncommutative) unital ring with no zero-divisors let and $\mathcal{N}$ a non-zero right module that is projective and torsion-free. Projectivity of course implies that $\mathcal{N}$ is flat, but does projectivity together with torsion-freeness suffice to imply that $\mathcal{N}$ is faithfully flat?