Deissler Rank Complexity of Powers of Indecomposable Injective Modules

Minimality ranks in the style of Deissler are one way of measuring the structural complexity of minimal extensions of first-order structures. In particular, positive Deissler rank measures the complexity of the injective envelope of a module as an extension of that module. In this paper we solve a problem of the second author by showing that certain injective envelopes have the maximum possible positive Deissler rank complexity. The proof shows that this complexity naturally reflects the internal structure of the injective extension in the form of the levels of the Matlis hierarchy. In this paper we present a general and positive solution to a problem raised in Kucera [4]. The problem concerns the structural complexity of injective modules over a commutative Noetherian ring . An injective module is essentially one wherein every formally consistent system of linear equations has a solution. The injective envelope of a module (minimal injective extension) can be constructed by adding solutions to linear systems, in a manner analogous to the construction of the algebraic closure of a field (see, for instance, Kucera [5] for details). Thus it makes sense to analyze the structural complexity of injective envelopes in terms of patterns of solutions of linear systems. There is further support for this idea from the viewpoint of mathematical logic. A positive primitive formula (in the first-order language of -modules) is a formula φ( x) of the form ∃ y n ∧