By Kolman, Busby, Ross

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**Estructuras de matemática discreta para la computación**

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If M is a finitely presented module and a is a tuple from M, then the pp-type of a in M is finitely generated. Proof. From any presentation of M and expression of a in terms of those generators we produce a generator for ppM (a). Suppose that b is a finite tuple of elements which together generate M and let bH = 0, where H is a matrix with entries in the ring, give a finite generating set for the relations on b. Let G be a matrix such that a = bG. Then the required I 0 pp condition is ∃y (x y) = 0.

Are pp-definable subgroups of M. For each λ ∈ k, so is the subgroup {a ∈ Me1 : β(a) = λα(a)}. 1. 5. Let D be a division ring. If M is any D-module, then End(M) acts transitively on the non-zero elements of M, so the only pp-definable subgroups of M are 0 and M. Sometimes we will use the following notation from model theory: if χ is a condition with free variables x, we write χ (x) if we wish to display these variables, and if a is a tuple of elements from the module M, then the notation M |= χ (a), read as “M satisfies χ (a)” or “a satisfies the condition χ in M”, means a ∈ χ (M), where χ (M) denotes the solution set of χ in M.

The simplest examples of such conditions θ are those of the form xr = 0 for some r ∈ R. In this case θ (M) = {a ∈ M : ar = 0} = annM (r), the annihilator of r in M. Indeed any condition of the type above may be regarded as the generalised annihilator condition xH = 0, where H is the n × m matrix (rij )ij . Then θ (M) is just the kernel of the map, x → xH from M n to M m which is defined by right multiplication by the matrix H ; a map of left End(M)-modules. Such annihilator-type conditions will not be enough: we close under projections to obtain generalised divisibility conditions.