= Split algebraic torus
{title2=$T\cong_k\mathbb G_m^r$}
A split algebraic torus over a <field> $k$ is an <algebraic group> isomorphic over $k$ to a product $\mathbb G_m^r$ of <multiplicative group schemes>. Its coordinate ring is the <Laurent polynomial ring>
$$
k[x_1^{\pm1},\ldots,x_r^{\pm1}],
$$
and its $k$-points are $(k^\times)^r$. Choosing such an isomorphism supplies torus coordinates; the torus need not come with preferred coordinates.
Every <algebraic torus character> is a monomial $x_1^{m_1}\cdots x_r^{m_r}$ with integer exponents. The character group is therefore $\mathbb Z^r$, and <cocharacters> form the dual lattice. This integral structure makes split tori the dense groups in ordinary fan-based <toric geometry>.
The adjective split concerns the field of definition, not connectedness of the real points. For example, $\mathbb G_m$ is split over $\mathbb R$ although $\mathbb R^\times$ has two connected components. The real <norm-one algebraic torus> from $\mathbb C/\mathbb R$ is connected as a real circle but is nonsplit. <Algebraic Groups, Lie Groups, and their Arithmetic Subgroups by James S. Milne>[ChapterI section14, tori and character modules]{ref}
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