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We study A, finite dimensional real division algebra with left unit e, satisfying: for all x ∈ A, (E1) (x, x, x) = 0, (E2) (x², x², x² ) = 0, (E3) x² e = x2 and (E4) (xe)e = x. We show that: • If A satisfies to (E1), then e is the unit element of A. • (E1) =⇒ (E2) =⇒ (E3) =⇒ (E4). In two-dimensional, we determine A satisfying (Ei)i∈{1,2,3,4}. We have A satisfies to (E1) (E2) (E3) (E4) A isomorphic to R; C R; C; ⋆C R; C; ⋆C R; C; ⋆C; L(1, −1, γ, 1) We show as well as (E1) =⇒ (E2) ⇐⇒ (E3) =⇒ (E4). We finally study the fused four-dimensional real division algebras satisfying (Ei)i∈{1,2}. We have shown that those which verify (E2) are H, ⋆H and C ⊕ B. and that H is the only fused algebra division with left unit satisfies to (E1). [ABSTRACT FROM AUTHOR]

Let R be an associative ring and M a unitary left R-module. An R-module M is said to be uniserial if its submodules are linearly ordered by inclusion. A serial module is a direct sum of uniserial modules. In this paper, we bring our modest contribution to the open problem listed in the book of Alberto Facchini "Module Theory" which states that: is any direct summand of a serial module serial? The answer is yes for particular rings and R-modules. [ABSTRACT FROM AUTHOR]