The main conclusions of this thesis can be summarized as follows:
• We have defined the notion of pure ad-nilpotent element in Chapter 2. The extended centroid plays an important role in this definition. It is a technical condition, since every ad-nilpotent element can be expressed as an orthogonal sum of pure ad-nilpotent elements of decreasing indices. Furthermore, this definition allows us to give a more precise description of such elements, and to weaken the torsion conditions required to the whole algebra. For more details we refer the reader to Section 2.1.
• We have proved that for any pure ad-nilpotent element a in a semiprime associative algebra R of index n with R free of and s-torsion, where , there exists λ in the extended centroid such that a – λ is nilpotent of index s. This fact is proved in the Theorem 2.2.4. We have weakened the conditions of Theorem [54, Theorem 1.3].
• Considering a semiprime associative algebra R with involution * we have described any skew-symmetric pure ad-nilpotent element a of index n depending on n modulo 4: If n ≡4 0 then the indexes of ad-nilpotence of a in R and K do not coincide and there exists a corner of R satisfying a PI. If n ≡4 1 then the indexes of ad-nilpotence of a in R and K coincide and there exists λ in the extended centroid such that a – λ is nilpotent. If n ≡4 3 then we can decompose a in an orthogonal sum a = a1 + a2 such that, if a1 ≠ 0, a1 is ad-nilpotent of R of index n (so there exists λ in the extended centroid such that a1 – λ is nilpotent) and, if a2 ≠ 0, a2 is ad-nilpotent of R of index n + 2 (therefore there exists a corner of R that satisfies a PI). The case n ≡4 2 cannot occur. This description has been proved in Theorem 2.3.6.
• In the same spirit as in the non-super setting, we have given descriptions of ad-nilpotent elements in prime associative superalgebras with superinvolution. These descriptions relate the index of ad-nilpotence of a homogeneous element with its nilpotence index. An important remark related with these descriptions is that every ad-nilpotent element has a minimal polinomial in the central closure with one root in the extended centroid. We refer readers to Theorems 3.1.2, 3.2.4, 3.2.5 for more details.
• To conclude our study about ad-nilpotent elements in associative algebras and superalgebras, in Chapter 4, we have given examples of elements appearing in these descriptions. The examples are matrices considered in the associative superalgebra M(r|s) over a field with a nontrivial superinvolution. Although we have considered a superalgebra, we also provide examples for descriptions of ad-nilpotent elements in associative algebras when we restrict the examples to the even part of the matrices M(r|s).
• For a Lie superalgebra L with an even ad-nilpotent element a of index 3 we have shown that ([a, [a, L]], ·) with a new product · defined by [a, [a, x]] · [a, [a, y]] := ½[a, [a, [x, [a, y]] is a Jordan superalgebra isomorphic to La = L(a)/KerL(a) where L(a) = (L, •) with x • y := [x, [a, y]] and KerL(a) := {x ∈ L | [a, [a, x]] = 0}. This result has been proved in Proposition 5.1.2.
• However, for a Lie superalgebra with an odd ad-nilpotent element the same construction gives a super anticommutative superalgebra, hence it cannot be a Jordan superalgebra. Instead, we have proved that it is possible to construct a Jordan superpair, see 5.2.3.
• Finally, we have defined the subquotient of a Lie superalgebra associated to an abelian inner ideal and we have proved that it is a Jordan superpair (Proposition 5.3.3). Moreover, we have shown that the subquotient corresponds to the construction made before (Corollary 5.3.5).
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