Primitive lattice vector ($\gcd (u_1,\dots ,u_n)=1$)

A primitive lattice vector is a nonzero element $u$ of a finite-rank free abelian group $N$ that cannot be written $u=kv$ with $v\in N$ and an integer $|k|>1$. In an integral basis, this is equivalent to the greatest common divisor of its coordinates being one. The property is independent of the chosen lattice basis.

For example, $(2,3)$ is primitive although neither coordinate is one: the integer functional $\ell (x,y)=-x+y$ satisfies $\ell (2,3)=1$. Hence $N=\mathbb{Z}u\oplus \ker \ell$, showing that $u$ extends to a lattice basis. In contrast, $(2,4)=2(1,2)$ is not primitive.

Every rational ray of a fan contains a unique primitive generator pointing along it. This generator gives the integral normalization used in toric divisors and cocharacters; replacing it by a larger positive multiple leaves the geometric ray unchanged but changes its lattice data.