In this paper, we define the notion of minimal prime ideals hoops and investigate some properties them. Then by using annihilators, study relation between annihilators. Also, introduce zero divisors elements prove that set all is a union hoop. Finally, notions maximal hoop, two new as p-ideal m-ideal. them every semi-simple hoop an m-ideal it.

In the last twenty years several methods for computing primary decompositions of ideals in multivariate polynomial rings over fields (Seidenberg (1974), Lazard (1985), Kredel (1987), Eisenbud et al. (1992)), the integers (Seidenberg, 1978), factorially closed principal ideal domains (Ayoub (1982), Gianni et al. (1988)) and more general rings (Seidenberg, 1984) have been proposed. A related prob...