Length functions, multiplicities and algebraic entropy

Abstract

We consider algebraic entropy defined using a general discrete length function L and will consider the resulting entropy in the setting of ROEX -modules. Then entropy will be viewed as a function h L on modules over the polynomial ring ROEX extending L. In this framework we obtain the main results of this paper, namely that under some mild conditions the induced entropy is additive, thus entropy becomes an operator from the length functions on R-modules to length functions on ROEX -modules. Furthermore, if one requires that the induced length function h L satisfies two very natural conditions, then this process is uniquely determined. When R is Noetherian, we will deduce that, in this setting, entropy coincides with the multiplicity symbol as conjectured by the second named author. As an application we show that if R is also commutative, the L-entropy of the right Bernoulli shift on the infinite direct product of a module of finite positive length has value 1, generalizing a result proved for Abelian groups by A. Giordano Bruno.

Intrinsic algebraic entropy

“…If G is torsion, it is well known (see [6] or [19]) that ent(φ) = h(φ), hence the second assertion follows. 2…”

A soft introduction to algebraic entropy

“…Indeed, in Proposition 3.1, we have seen that two K [X ]-modules V φ and W ψ are isomorphic exactly when φ and ψ are conjugated, which implies that isomorphic K [X ]-modules have the same algebraic entropy. Therefore, ent can be viewed as an invariant of the category Mod(K [X ]) with values in R ≥0 ∪ {∞} (this point of view is fully developed in [15]). Moreover, as every module is the direct limit of its finitely generated submodules, Proposition 3.3 says in particular that the algebraic entropy is an upper continuous invariant of…”

“…Very recently, using ideas from both the above proofs and borrowing techniques typical of p-groups and of torsion-free groups, a very general Addition Theorem has been proved in [15] for suitable subcategories of modules over arbitrary rings, dealing with the algebraic entropies associated with length functions (see their definition below in this section).…”

“…Nowadays, this theory is extended to R-modules over arbitrary rings R (see [2,15,16,18,21]), and also to topological groups (see [9]). As usual, the more general the considered ring R is, the less specific are the results obtainable for R-modules.…”

The Mal'cev Rank for Modules over Pseudo-Valuation Domains

“…This notion, introduced by Northcott and Reufel ( [14]) as a generalization of the classical Jordan-Hölder length of modules, was also studied by Vámos ([18]), and extended by Ribenboim ([15]) to maps with values into an ordered abelian group. Further investigation on length functions have been made very recently by Salce, Virili, Vámos [16], Fusacchia [9], and, taking a different point of view, Zanardo [20].…”