Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetry in theoretical physics.
In doing so one moves from fields to commutative superalgebras and from vector spaces to modules.
A right supermodule over A is a right module E over A with a direct sum decomposition (as an abelian group) such that multiplication by elements of A satisfies for all i and j in Z2.
If A is supercommutative, then every left or right supermodule over A may be regarded as a superbimodule by setting for homogeneous elements a ∈ A and x ∈ E, and extending by linearity.
For supermodules E and F, let Hom(E, F) denote the space of all right A-linear maps (i.e. all module homomorphisms from E to F considered as ungraded right A-modules).