support and hospitality while most of this work was done during a membership in the Spring of 1997.

Contragredient modules and some related issues
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This section is preliminary. Here we review some basic notions and facts, including the notion of contragredient module and an extension of this. We shall discuss the notion of right module and prove certain results.

We shall use standard definitions and notions such as the notions of vertex operator algebra, (irreducible) module, homomorphism, for which we refer the reader to \[FHL\] and \[FLM\]. Since we here only deal with the representation theory of vertex operator algebras, we fix a vertex operator algebra $V$ throughout this paper. We shall typically use $x,y, x_{1},x_{2}, x_{3},\dots$ for mutually commuting formal variables, use letters $u,v$ for elements of $V$ and use $w, w_{(1)},w_{(2)},\dots$ for elements of $V$-modules. We use ${\Z}$ for the set of all integers, ${\N}$ for the set of all nonnegative integers and ${\C}$ for the field of complex numbers. We shall also use certain relatively new notions, which we recall next.

[*Throughout this paper, a $V$-module always stands for an ordinary $V$-module unless it is labeled as a weak or generalized $V$-module.*]{}

Vertex operator algebra $V$ 