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Abstract: We study semiprime superalgebras with superinvolution, under certain additional regularity conditions. More precisely, we assume regularity conditions either on every nonzero homogeneous symmetric element, or on every nonzero homogeneous skewsymmetric element. [Copyright &y& Elsevier]

In this note we emphasise the relationship between the structure of an associative superalgebra with superinvolution and the structure of the Lie substructure of skewsymmetric elements. More explicitly, we show that if A is a semiprime associative superalgebra with superinvolution and K is the Lie superalgebra of skewsymmetric elements satisfying [K2, K2] = 0, then A is a subdirect product of orders in simple superalgebras each at most 4-dimensional over its center. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate the Lie structure of the Lie superalgebra K of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U is a Lie ideal of K, then either there exists an ideal J of A such that the Lie ideal is nonzero and contained in U, or A is a subdirect sum of , , where the image of U in is central, and is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center. [Copyright &y& Elsevier]

The maximal subalgebras of the finite dimensional central simple associative superalgebras, possibly endowed with a superinvolution, are determined. This relies on the corresponding description by M. Racine in the ungraded case, which is completed here too. [Copyright &y& Elsevier]