The idea of a sutured topological quantum field theory was introduced by
Honda, Kazez and Mati\'c (2008). A sutured TQFT associates a group to each
sutured surface and an element of this group to each dividing set on this
surface. The notion was originally introduced to talk about contact invariants
in Sutured Floer Homology. We provide an elementary example of a sutured TQFT,
which comes from taking exterior algebras of certain singular homology groups.
We show that this sutured TQFT coincides with that of Honda et al. using
Z2-coefficients. The groups in our theory, being exterior algebras,
naturally come with the structure of a ring with unit. We give an application
of this ring structure to understanding tight contact structures on solid tori