= Torus factor
A torus factor of a <toric variety> is a product factor $\mathbb G_m^r$ in a toric decomposition $X\cong Y\times\mathbb G_m^r$. A nontrivial factor has $r>0$. This is stronger than merely containing a dense torus, which every toric variety does.
For a <toric fan> $\Sigma$ in a lattice $N$, let $N'=N\cap\operatorname{span}_{\mathbb R}(\bigcup_{\sigma\in\Sigma}\sigma)$. It is a saturated sublattice. Choosing a complementary lattice $N''$ gives
$$
X_\Sigma\cong X_{\Sigma\text{ in }N'}\times\mathbb G_m^{\operatorname{rank}N''}.
$$
The complementary splitting need not be canonical. Thus a proper linear span of the whole fan supplies a nontrivial torus factor.
For the fan consisting of one coordinate ray and zero in $\mathbb Z^2$, the affine coordinate ring is $k[x,y,y^{-1}]$, giving $\mathbb A^1\times\mathbb G_m$. This variety has no torus-fixed point because the second multiplicative coordinate cannot be fixed by every scaling. A fan with only lower-dimensional cones can still span the whole lattice, so absence of full-dimensional individual cones alone does not imply a torus factor.
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