Modules over linear spaces admitting a multiplicative basis

Abstract

Let V and W be two vector spaces over a base field F. It is said that V is a module over W if it is endowed with a bilinear map V x W → V, (v,w) ↦ vw. A basis B = {v_i}_i∈I of V is called multiplicative with respect to the basis B' = {w_j}_j∈J of W if for any i ∈ I, j ∈ J we have either v_i w_j = 0 or 0 ≠ v_i w_j ∈ Fv_k for some K ∈ I. We show that if V admits a multiplicative basis in the above sense then it decomposes as the direct sum V = ⊕_a V_a of well-described submodules admitting each one a multiplicative basis. Also, under a mild condition, the minimality of V is characterized in terms of the multiplicative basis and it is shown that the above direct sum is by means of the family of its minimal submodules, admitting each one a multiplicative basis. Some applications to the structure theory of arbitrary algebras with multiplicative bases, arbitrary algebraic pairs with multiplicative bases and modules over arbitrary algebras with multiplicative bases are also provided.