  We develop an elementary theory of partially additive rings as a foundation
of ${\mathbb F}_1$-geometry. Our approach is so concrete that an analog of
classical algebraic geometry is established very straightforwardly. As
applications, (1) we construct a kind of group scheme ${\mathbb GL}_n$ whose
value at a commutative ring $R$ is the group of $n\times n$ invertible matrices
over $R$ and at ${\mathbb F}_1$ is the $n$-th symmetric group, and (2) we
construct a projective space $\mathbb P^n$ as a kind of scheme and count the
number of points of ${\mathbb P}^n({\mathbb F}_q)$ for $q=1$ or $q=p^n$ a power
of a rational prime, then we explain a reason of number 1 in the subscript of
${\mathbb F}_1$ even though it has two elements.
