Let \(R\) be a commutative ring and \(X\) an \(R\)-module, and let \(F \coloneqq (\wait) \tensor_R X\) on \(R\)-modules.
\(F\) is right exact.
\(F\) preserves injections, that is, \(T \tensor \id_X\colon V \tensor_R X \to W \tensor_R X\) is injective for every injective \(R\)-linear map \(T\colon V \injects W\), if and only if \(X\) is flat, and in that case \(F\) is exact.
If \(R = k\) is a field, every \(k\)-module is flat, so \(T \tensor \id_X\colon V \tensor_k X \injects W \tensor_k X\) is injective for every injective \(k\)-linear map \(T\colon V \injects W\) and every \(k\)-vector space \(X\).
Over \(\ZZ\), tensoring does not preserve injections in general: \(\ZZ/2\ZZ\) is not a flat \(\ZZ\)-module. The map \(\ZZ \mapsvia{2} \ZZ\) is injective, and tensoring it with \(\ZZ/2\ZZ\) gives multiplication by \(2\) on \(\ZZ/2\ZZ\), which is the zero map and not injective.