International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-3, Issue-6, Jun- 2017] https://dx.doi.org/10.24001/ijaems.3.6.4 ISSN: 2454-1311 www.ijaems.com Page | 642 Certain Generalized Prime elements C. S. Manjarekar, A. N. Chavan Shivaji University, Kolhapur, India Abstract — In this paper we study different generalizations of prime elements and prove certain properties of these elements. Keywords— Prime, primary elements,weakly prime elements,weakly primary elements, 2-absorbing, 2-potent elements Math. Subject Classification Number:- 06F10, 06E20, 06E99. I. INTRODUCTION A multiplicative lattice L is a complete lattice provided with commutative, associative and join distributive multiplication in which the largest element 1 acts as a multiplicative identity. An element a ∈ L is called proper if a < 1. A proper element p of L is said to be prime if ab ˄ p implies a ˄ p or b ˄ p. If a∈ L, b ∈ L, (a : b) is the join of all elements c in L such that cb ˄ a. A properelement p of L is said to be primary if ab ˄ p implies a ˄ p or ௡ ˄ p for some positive integer n. If a ∈ L, then √ = ޕ{x ∈ ∗ / ݔ ௡ ˄ a, n ∈ Z+}. An element a ∈ L is called a radical element ifa = √ . An element a∈ L is called compact if a ≤⋁    implies a ≤  భ ˅  మ ˅… ˅   for some finite subset { ଵ,  ଶ …  ௡ }. Throughout this paper, L denotes a compactly generated multiplicative lattice with 1 compact in which every finite product of compact element is compact. We shall denote by  ∗ , the set of compact elements of L. An element i ∈ L is called 2-absorbing element if abc ˄i impliesab˄i or bc˄i or ca˄i. A proper element i∈ L is called 2-absorbing primary if for all a, b, c ∈ L, abc˄i implies either ab˄i or bc˄√ or ca˄√ . This concept was defined by U.Tekir et.al. in [7]. It is observed that every prime element is 2-absorbing. An element i∈ L is called semi-prime if i =√ . An element is i called 2-potent prime if ab ˄ ଶ implies a ˄i or b ˄i. (See [6]). Every 2-absorbing element of L is a 2-absorbing primary element of L. But the converse is not true. The element q = (12) is a 2-absorbing primary element of L but not 2-absorbing element of L. Also every primary element of L is a 2 absorbing primary element. But the converse is not true. The element q = (6) is a 2 absorbing primary element of L but not a primary element of L, since L is lattice of ideals of the ring. R =< Z,+, .>. For all these definition one can refer [1],[4],[5]. II. PRIME AND PRIMARY ABSORBING ELEMENTS The concept of primary 2-absorbing ideals was introduced by Tessema et.al. [5]. We generalize this concept for multiplicative lattices. An element i ∈ L is said to be weakly prime if 0 ≠ ab˄i implies a ˄i or b ˄i. It is easy to show that every prime element is 2- absorbing. Ex. The following table shows multiplication of elements in themultiplicative lattice L = 0, p, q, 1. In the above diagram 0, p, q are 2-absorbing. The concept of 2-absorbing primary ideals is defined by A. Badawi,U. Tekir, E. Yetkin in [6]. The concept was generalized for multiplicative lattices by F. Calliap, E. Yetkin, and U. Tekir [8]. Weslightly modified this concept and defined primary 2-absorbing element. Def.(2.1) An element i ∈ L is said to be weakly 2-absorbing if0 ≠ abc˄i implies ab˄i or bc˄i or ca˄i. (See [7]). Def.(2.2) An element i of L is called primary 2-absorbing ifabc˄i implies ab˄i or bc ˄√ or ca˄√ , for all a, b, c ∈ L. Ex. Every 2- absorbing element of L is primary 2- absorbing. We obtain now the relation between primary 2-absorbing element and 2-absorbing element. Theorem (2.3) Ifi is semi-prime and primary 2-absorbing element of a lattice L, then i is 2-absorbing. Proof:- Suppose i is primary 2-absorbing. Let abc˄i. Then ab˄i or bc˄√ or ca˄√ where i= √ . Therefore i is 2- absorbing. _ Theorem(2.4) If i is semi-prime and 2-potent prime element of Lthen i is prime. Proof:- Let ab˄i. Then ሺሻ ଶ =  ଶ  ଶ ˄ ଶ . Then  ଶ ˄i or  ଶ ˄i,since i is 2-potent prime. This implies that a ˄√ or b