Strongly Irreducible Submodules of Modules

On the Notion of Strong Irreducibility and Its Dual

“…An upper semilattice (L, ∨) is defined analogously asking that any two elements have a least upper bound a ∨ b. For any a, b ∈ L we set the interval of a and b to be the subset Strongly irreducible ideals and submodules have been studied in [5,6,15,18]. The dual notion of a strongly irreducible submodule was termed strongly hollow in [1] and our purpose is to use lattice theory to obtain unifying results on strongly irreducible elements either in the lattice of one-sided or two-sided ideals, submodules or in the dual lattices of those.…”

Discussions on Necessary Conditions for Equivalency of Primary, Irreducible, and Strongly Irreducible Submodules

“…Proposition 14: Let R be an UFD and M be a multiplication faithful module and N a M  be a submodule of M. Then, N is SI iff N is primary, [3].…”

“…Proposition 1: Let M be a multiplication and N be a prime submodule of M. Then, N is a SI submodule, [3].…”

“…The concept was generalized at 2006 by Khaksari [3]. In the present paper, [4], the relationship between the irreducible, strongly irreducible, prime, and primary submodules of an R-modules is investigated.…”

“…Proposition 11: Let R be an UFD and I be an ideal of R and x, y  R, then I is a SI ideal iff [x, y]  I which implies x  I or y  I -where [x, y] represents Least Common Multiple (LCM) of x and y - [3].…”