  We introduce tropical dual numbers as an extension of tropical semiring. By
this innovation, one can work with honest ideals, instead of congruences, and
recover the Euclidean topology on affine tropical spaces similar to Zariski's
approach in classical algebraic geometry. The bend loci of an ideal over
tropical dual numbers coincides with the bend loci of a congruence over
tropical semiring and this enables tropical dual numbers to serve as an
algebraic structure for tropical geometry. Tropical Zariski topology on affine
tropical spaces whose closed sets are non-linear loci of ideals over tropical
dual numbers offers an alternative point of view to strong Zariski topology
defined by Giansiracusa.
