Modules are to rings what vector spaces are to fields: the same axioms, applied to a ring R instead of a field F, yield modules. The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. For example, modules need not have bases, as the Z-module (that is, abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field, with the elements being called vectors. Some authors use the term vector space to mean modules over a division ring. The algebro-geometric interpretation of commutative rings via their spectrum allows the development of concepts such as locally free modules, the algebraic counterpart to vector bundles. Affine and projective spaces Roughly, affine spaces are vector spaces whose origins are not specified. More precisely, an affine space is a set with a free transitive vector space action. In particular, a vector space is an affine space over itself, by the map If W is a vector space, then an affine subspace is a subset of W obtained by translating a linear subspace V by a fixed vector ; this space is denoted by (it is a coset of V in W) and consists of all vectors of the form for An important example is the space of solutions of a system of inhomogeneous linear equations generalizing the homogeneous case above, which can be found by setting in this equation. The space of solutions is the affine subspace where x is a particular solution of the equation, and V is the space of solutions of the homogeneous equation (the nullspace of A). The set of one-dimensional subspaces of a fixed finite-dimensional vector space V is known as projective space; it may be used to formalize the idea of parallel lines intersecting at infinity. Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension k and flags of subspaces, respectively. Related concepts Specific vectors in a vector space Zero vector (sometimes also called null vector and denoted by ), the additive identity in a vector space. In a normed vector space, it is the unique vector of norm zero. In a Euclidean vector space, it is the unique vector of length zero. Basis vector, an