Let $\Lambda := \text{Span}_{\mathbb Z} \{v_1,\dots, v_k\} \le \mathbb Z^d$ ($k\le d$) with each vector $v_i$ primitive (namely components are coprime integers).
Is it true that
$$\text{Span}_{\mathbb R} \{v_1,\dots, v_k\} \cap \mathbb Z^d \subset \Lambda ?$$
This looks correct to me, but unfortunately I don't know how to verify this purely algebraically.
Let $v_1=(1,1),v_2=(1,-1)\in\mathbb R^2$, so that $\Lambda=\{(x,y)\in\mathbb Z^2:x\equiv y\pmod2\}$. Then, $Span_{\mathbb R}(v_1,v_2)=\mathbb R^2$ so $$Span_{\mathbb R}(v_1,v_2)\cap\mathbb Z^2=\mathbb Z^2\not\subset\Lambda.$$