The author develops the theory of left and right ideals, in finite-dimensional Grassmann algebras, as their annihilators to a broad extent. Left and right duality operators mapping left and right ideals onto another are used in studying the annihilators of the ideals. The main result states, that the left ideal \({\mathfrak I}\) annihilates the right ideal \(J\) and vice versa, if the dual of the left ideal \({\mathfrak I}\) corresponds to the right ideal \(J\). A further duality is given for two-sided ideals, where the two dual ideals are the annihilators of each other. The dual ideal of the principal ideal of a 2-form is completely determined. The result enables one to give a condition for the factorization of Grassmann elements using duality. An example is given.