This study aims to investigate free objects in the category of acts over an arbitrary semigroup S. We consider two generalizations of free acts over arbitrary semigroups, namely acts with conditions (F1) and (F2), and give some new results about (minimal) PRIME SUBACTs and radical SUBACTs of any S-act with condition (F1). Furthermore, some lattice structures for some collections of radical SUBACTs of free S-acts are introduced. We also obtain some results about the relationship between radical SUBACTs of free S-acts and radical ideals of S. Moreover, for any PRIME ideal P of a semigroup S with a zero, we find a one-to-one correspondence between the collections of P-PRIME SUBACTs of any two free S-acts. Also, it is shown that all free S-acts have isomorphic lattices of radical multiplication SUBACTs.

Let R be a commutative ring with identity. A proper ideal P of R is an (n - 1, n)-Fm-PRIME ((n - 1, n)-weakly PRIME) ideal if a1, … , anÎR, a1 … anÎP\P m (a1 … anÎP\{0}) implies a1 … ai-1 ai+1 … anÎP, for some iÎ{1, …, n}; (m, n³2).In this paper several results concerning (n - 1, n)-Fm-PRIME and (n - 1, n)-weakly PRIME ideals are proved. We show that in a Noetherian domain a Fm-PRIME ideal is primary and we show that in some well-known rings (n - 1, n)-Fm-PRIME ideals and (n - 1, n)-PRIME ideals coincide.

In this paper, by considering the notions of maximal, PRIME, and minimal PRIME filters in residual lattices, we study their properties and some examples of residuated lattices are given. It is reminded first the notion of a filter of a residuated lattice and then we show that each proper filter is contained in a maximal filter and therefore in a PRIME filter. We investigate the PRIME filters in an MTL algebra and state and prove the fundamental theorem of the PRIME filters for residuated lattices. We investigate the residuated lattices in which each its filter is principal, and by using the PRIME filters, we inspect co-annihilators. Then the notion of a minimal PRIME filter in a residuated lattice is introduced and a fundamental theorem for minimal PRIME filters in residuated lattice is stated and proved. Finally, we introduce the divisor filters in a residuated lattice and by using them,we investigate the minimal PRIME filters.

Let R be a commutative ring with identity. We give a new generalization to PRIME ideals called ,-PRIME ideal. A proper ideal P of R is called an ,-PRIME ideal if for all a,b in R with ab 2 P, then a 2 P or , (b) 2 P where ,2 End(R). We study some properties of ,-PRIME ideals analogous to PRIME ideals. We give some characterizations for such generalization and we prove that the intersection of all ,-PRIMEs in a ring R is the set of all ,-nilpotent elements in R. Finally, we give new versions of some famous theorems about PRIME ideals including ,-integral domains and ,-, elds.