Abstract
Conditions that are necessary for the relative annihilator in lower BCK-semilattices to be a prime ideal are discussed. Given the minimal prime decomposition of an ideal A, a condition for any prime ideal to be one of the minimal prime factors of A is provided. Homomorphic image and pre-image of the minimal prime decomposition of an ideal are considered. Using a semi-prime closure operation "cl", we show that every minimal prime factor of a cl-closed ideal A is also cl-closed.