Simple Special Jordan Superalgebras with Associative Nil-Semisimple Even Part

We describe unital simple special Jordan superalgebras with associative nil-semisimple even part. In every such superalgebra $J=A+M$, either $M$ is an associative and commutative $A$-module, or the associator space $(A,A,M)$ coincides with $M$. In the former case, if $J$ is not a superalgebra of the non-degenerate bilinear superform then its even part $A$ is a differentiably simple algebra and its odd part $M$ is a finitely generated projective $A$-module of rank 1. Multiplication in $M$ is defined by fixed finite sets of derivations and elements of $A$. If, in addition, $M$ is one-generated then the initial superalgebra is a twisted superalgebra of vector type. The condition of being one-generated for $M$ is satisfied, for instance, if $A$ is local or isomorphic to a polynomial algebra. We also give a description of superalgebras for which $(A,A,M)\neq 0$ and $M\cap [A,M]\ne0$, where $[\, ,\,]$ is a commutator in the associative enveloping superalgebra of $J$. It is shown that such each infinite-dimensional superalgebra may be obtained from a simple Jordan superalgebra whose odd part is an associative module over the even.