A module M is called square-free if it contains nonon-zero omorphic submodules and B with A∩B= 0. Dually, Mis dual-square-free has no proper M=A+B M/A∼=M/B. In this paper we show that M=⊕i∈I Mi, then iff each Mi Mj ⊕j=i∈I are orthogonal. M=⊕ni=1Mi, dual-square-free, 1⩽i⩽n, ⊕ni=jMi factor-orthogonal. Moreover, in the finite case, weshow M=⊕i∈ISi a direct sum of non-is simple modules, dual-square...

In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here the fact that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the ...

Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem  2.8}). We shall determine the structure of special elements (which are introduced after  Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of ...