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In this paper, we define the direct sum relation on a module and prove that this is a partial order when the module is regular. We also focus on a regular support and a strong regular support of an element in a module, which are extensions of a generalized inverse and a reflexive generalized inverse of an element in a ring, respectively. Using regular supports and strong regular supports, we present various characterizations of the direct sum order as an application. We obtain some decompositions of a module via the direct sum order.

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].

Let [Formula: see text] be an arbitrary ring with identity, [Formula: see text] a right [Formula: see text]-module and [Formula: see text] a fully invariant submodule of [Formula: see text]. The notion of an [Formula: see text]-inverse split module [Formula: see text] has been defined and studied by the present authors recently. In this paper, we introduce its dual notion, namely, dual [Formula: see text]-inverse split module [Formula: see text]. This work is devoted to investigation of various properties and characterizations of a dual [Formula: see text]-inverse split module [Formula: see text]. We include applications for rings and cosingular submodules. We also deal with the notion of relatively dual inverse splitness to investigate direct sums of dual inverse split modules.

Let [Formula: see text] be an arbitrary ring with identity and [Formula: see text] a right [Formula: see text]-module with the ring [Formula: see text] End[Formula: see text] of endomorphisms of [Formula: see text]. The notion of an [Formula: see text]-inverse split module [Formula: see text], where [Formula: see text] is a fully invariant submodule of [Formula: see text], is defined and studied by the present authors. This concept produces Rickart submodules of modules in the sense of Lee, Rizvi and Roman. In this paper, we consider the submodule [Formula: see text] of [Formula: see text] as [Formula: see text] and [Formula: see text], and investigate some properties of [Formula: see text]-inverse split modules and [Formula: see text]-inverse split modules [Formula: see text]. Results are applied to characterize rings [Formula: see text] for which every free (projective) right [Formula: see text]-module [Formula: see text] is [Formula: see text]-inverse split for the preradicals such as [Formula: see text] and [Formula: see text].