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In this paper, we define the concepts of the radical of an ideal and a primary ideal in posets. Further, the analogue of the first and the second uniqueness theorems regarding primary decomposition of an ideal are obtained. In the last section, we prove that if an ideal in a poset Q has a minimal primary decomposition, then the diameter of the corresponding zero-divisor graph with respect to this ideal is exactly equal to three.

In the first section of this paper, we prove an analogue of Stone’s Theorem for posets satisfying DCC by using semiprime ideals. We also prove the existence of prime ideals in atomic posets in which atoms are dually distributive. Further, it is proved that every maximal non-dense (non-principal) ideal of a 0-distributive poset (meet-semilattice) is prime. The second section focuses on the characterizations of (minimal) prime ideals in pseudocomplemented posets. The third section deals with the generalization of the classical theorem of Nachbin. In fact, we prove that a dually atomic pseudocomplemented, 1-distributive poset is complemented if and only if the poset of prime ideals is unordered. In the last section, we have characterized 0-distributive posets by means of prime ideals and minimal prime ideals.

In this paper, we study Baer ideals in posets and obtain some characterizations of Baer ideals in [Formula: see text]-distributive posets. Further, we prove that in an ideal-distributive poset, every ideal is Baer (normal) if and only if every prime ideal is Baer (normal). We extend the concept of a quasicomplement to posets and prove characterizations of quasicomplemented poset. This extend the results regarding Baer ideals and quasicomplemented lattices mentioned in [Y. S. Pawar and S. S. Khopade, [Formula: see text]-ideals and annihilator ideals in 0-distributive lattices, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 49(1) (2010) 63–74; Y. S. Pawar and D. N. Mane, [Formula: see text]-ideals in 0-distributive semilattices and 0-distributive lattices, Indian J. Pure Appl. Math. 24(7–8) (1993) 435–443] to posets.