Algebraic entropy for valuation domains

Abstract Let R be a non-discrete Archimedean valuation domain, G an R-module, Φ ∈ EndR(G).We compute the algebraic entropy entv(Φ), when Φ is restricted to a cyclic trajectory in G. We derive a special case of the Addition Theorem for entv, that is proved directly, without using the deep results and the difficult techniques of the paper by Salce and Virili [8].


Introduction
A sketchy definition of the concept of algebraic entropy for endomorphisms of Abelian groups was given in a 1965 article by Adler-Konheim-McAndrew [1], dedicated to topological entropy. In 1975 this concept was resumed and developed by Weiss [14], in a paper that related algebraic and topological entropies using Pontryagin duality. In the algebraic context, as well as in other areas of mathematics and physics, entropy is viewed as a measure of the "average disorder" created by a transformation when we repeatedly apply it. Following this philosophy, in [4] Dikranjan, Goldsmith, Salce and Zanardo thoroughly investigated the algebraic entropy of [1] and [14]. In [4] the notion was mainly used to get a better understanding of endomorphism rings of Abelian p-groups. Thereafter, Salce and Zanardo [10] defined the algebraic entropy for R a commutative ring, and used the rank as an invariant to deal with the case of torsion-free Abelian groups. Many more papers devoted to algebraic entropy in its different aspects have appeared since; for an ample list of references, see, for instance, [3].
A central result for an algebraic entropy (however defined) is the Addition Theorem. It states that, given an endomorphism ϕ of a left R-module M (R not necessarily commutative), and a ϕ-invariant submodule N of M, the following formula holds ent(ϕ) = ent(ϕ| N ) + ent(φ) whereφ is the endomorphism of M/N induced by ϕ. Starting with results in [4] and [10], Salce, Vámos and Virili [7] proved that every algebraic entropy induced by a discrete length function satisfies the Addition Theorem (see the preliminaries for the unexplained notions).
The present note is mainly motivated by a recent paper of Salce and Virili [8], where the authors proved in full generality the Addition Theorem for the algebraic entropy induced by any length function, either nondiscrete or discrete. So, this important new result extends and completes the above recalled one of [7], that was proved using methods applicable only to discrete length functions.
Northcott and Reufel [6] defined a length function for modules over a valuation domain R. In the case where R is Archimedean non-discrete, i.e., its value group is a dense ordered subgroup of R, we get the basic example of a non-discrete length function. In the present paper, we in fact consider a valuation domain R of this type, the length function Lv determined by the valuation v of R, and the algebraic entropy entv induced by Lv.
Let G be an R-module, ϕ ∈ End R (G). Our main purpose is to compute entv(ϕ), when ϕ is restricted to a cyclic trajectory in G (see the definitions in the first section). Of course, this computation, made in Theorem 4.5, may be helpful to determine entv(ϕ) on all of G. It is worth noting that a thorough knowledge of finitely generated modules over valuation domains is needed to get this result. We remark that Proposition 4.4 was used in [8] to prove the Uniqueness Theorem 7.3.
From Theorem 4.5 we also derive a special case of the Addition Theorem for entv, namely Proposition 4.7, that is proved directly, without using the deep results and the difficult techniques of [8].

Preliminaries and definitions.
For the notions and basic results on the theory of valuation domains and their modules, in particular the finitely generated ones, we refer to the book by Fuchs and Salce [5].
A general notion of length function was introduced by Northcott and Reufel [6]. This concept was investigated by Vámos [11], [12], by the author [15], and by Virili [13], in the context of Grothendieck categories. We recall below the definition.
Let R be any ring. A length function L on Mod R is a map L : (2) L is upper continuous, that is, for any M ∈ Mod R, L(M) is the sup of the lengths L(N), where N ranges among the finitely generated submodules of M.
L is said to be discrete if its image (without ∞) form an ordered semigroup of R ≥0 isomorphic to N; otherwise, L is called non-discrete.
In what follows, R is an Archimedean valuation domain, i.e., the value group Γv of R is an ordered subgroup of R. We denote by Q the field of fractions of R, by v the valuation on Q determined by R, and by P the maximal ideal of R, i.e., P = {x ∈ Q : v(x) > 0}. Of course, under the present circumstances, P is the unique nonzero prime ideal of R. We confine ourselves to non-discrete valuations, i.e., Γv is a dense subgroup of R. If I is an ideal of the valuation domain R, we set v(I) = inf{v(r) : r ∈ I}.
A submodule N of the R-module M is said to be pure if N ∩ rM = rN for every r ∈ R (see [5], Ch.I). For M an R-module, we denote by Ann(M) its annihilator. We define gen M to be the minimal cardinality of a generating system of M, when M is finitely generated; otherwise, we set gen M = ∞.
The next results on finitely generated modules over a valuation domain R were firstly proved in [9]; a neat discussion on this subject may be found in [5], Ch.V.5. Let X be a finitely generated R-module; then gen X = dim R/P (X/PX). Say gen X = n, and let {x 1 , . . . , xn} be a generating set for X. Then there exists a reordering z i = x τ(i) of the generators x i (τ a suitable permutation of {1, . . . , n}) such that, setting Z 0 = 0 and Z i = ⟨z 1 , . . . z i ⟩, for 1 ≤ i ≤ n, the following properties are satisfied (a) each Z i is pure in X; The above sequence of ideals is determined by X, and is called the annihilator sequence of X. Note that A i = Ann(z i + Z i−1 ). We say that 0 = Z 0 < Z 1 < · · · < Z n−1 < Zn = X is a pure-composition series of X.
Now we recall some basic facts proved in [6] (see also [15]).  (2) If X is finitely generated with annihilator sequence A 1 , . . . , An, then Lv(X) = ∑︀ n i=1 v(A i ). Since we assume that Γv is a dense subgroup of R, it is clear that Lv is not discrete. Let us observe that Lv(M) = ∞ whenever M is not a torsion module: in fact, in that case M contains a copy of R, hence ∞ = Lv(R) ≤ Lv(M) ≤ ∞. It follows that Lv is not significant outside the category of the torsion R-modules.
We recall now the usual definition of algebraic entropy induced by a length function, applying it to Lv (cf. [10]).
Let G be an R-module and denote by fL(G) the family of its submodules N such that Lv(N) is finite (we just say that N has finite length). If ϕ ∈ End R (G), for every positive integer n and every F ∈ fL(G) we set will be called the ϕ-trajectory of F. The ϕ-trajectory of an element x is just the ϕ-trajectory of the cyclic submodule Rx, and, obviously, it coincides with the smallest ϕ-invariant submodule of G containing x. It is simply denoted by T(ϕ, x), and called the cyclic ϕ-trajectory generated by x.
Given the submodule F of finite length and the endomorphism ϕ of G, for each n ≥ 1 we take the real number and define Since Lv is a length function, the above limit exists and is finite. We define the algebraic entropy induced by Lv (algebraic v-entropy, for short) of an endomorphism ϕ of G as entv(ϕ) = sup F∈fL(G) H(ϕ, F).
Property (2) says that every finitely generated torsion R-module has finite length. The converse is not true when the valuation v is non-discrete. Indeed, under the present circumstances we have v(P) = 0, hence, by the definition, Lv(R/P) = 0, and therefore any R/P-vector space V has length 0, not just the finitely generated ones. Less obvious examples of R-modules not finitely generated but with finite length may be found in Example 2.2 of [10] (see also Example 4.5 of [8]).
Naturally, it should be desirable to compute the algebraic entropy taking not all the submodules of finite length, but just the finitely generated ones. This is possible by Proposition 4.2 of [8], proved for any ring R and any length function L of Mod R. We restate that result just for the case we are interested in.

Proposition 2.1. ([8]) In the above notation, if G is a torsion R-module and ϕ ∈ End
where F(G) denotes the set of the finitely generated submodules of G.
In what follows, we will always compute the entropy confining ourselves to finitely generated torsion submodules. Indeed, if M contains a torsion-free finitely generated submodule, then entv(ϕ) = ∞ for every 0 ≠ ϕ ∈ End R (G). Note also that entv(ϕ) = 0 for every 0 ≠ ϕ ∈ End R (N), when N is a finitely generated torsion R-module: indeed, in this case ϕ is annihilated by a monic polynomial with coefficients in R, and therefore T(ϕ, N) is finitely generated (cf. the proof of Proposition 4.1).
We end this section observing that entv satisfies the standard basic properties of every algebraic entropy induced by a length function (e.g., see [10]). We don't recall those properties, since we won't need them.

Computation of v-entropies.
The aim of this section is to get a good description of the cyclic ϕ-trajectories contained in the torsion Rmodule G, where ϕ ∈ End R (G).
To simplify the notation, in what follows, for assigned ϕ ∈ End R (G) and x ∈ G, we will denote by Tn the n-partial trajectory Tn(ϕ, x).
Proof. Let us assume, for a contradiction, that gen Tm < m for some m > 0. Then ∑︀ m−1 i=0 a i ϕ i x = 0, for suitable a i ∈ R not all in P. From this relation it follows that there exist polynomials p 0 , p 1 , with p 0 monic, such that p 0 (ϕ)x = λp 1 (ϕ)x, where λ ∈ P. Now take k > 0 large enough such that λ k x = 0. This is possible, since R is archimedean. Then we get p k is a monic polynomial, of degree h, say, and so Tn ⊆ T h+1 for every n > 0. It follows that T(ϕ, x) = T h+1 is finitely generated, impossible.
The following two lemmas are crucial to get the next Proposition 4.4.
for some m > 0, a i ∈ R and λ ∈ P, then y = 0.
Proof. We assume, for a contradiction, that y ≠ 0, hence 0 ≠ Ann y ≠ R. Since R is archimedean, there exists r ∉ Ann y such that λry = 0. Then we get We reached a contradiction. Lemma 3.3. Let G be a torsion R-module, ϕ ∈ End R (G), y ∈ G. If ϕ k y = ∑︀ 0≤i<k a i ϕ i y + λ ∑︀ j>k b j ϕ j y, for some a i , b j ∈ R (almost all zero) and λ ∈ P, then ϕ k y = ∑︀ 0≤i<k c i ϕ i y, for suitable c i ∈ R.
Proof. Firstly we prove that, for every n ≥ k, we can write a relation for suitable a ni , b nj ∈ R, almost all zero. We make induction on n. The hypothesis yields the case n = k. Assume that the above relation is valid. Then we get for suitable a n+1,i , b n+1,j ∈ R, almost all zero. Now we prove that, for every m > 0, we can write a relation for suitable c mi , dmn ∈ R, almost all zero. We make induction on m. The case m = 1 is covered by the hypothesis. Assume that the above relation is valid. Since ϕ n y = ∑︀ 0≤i<k a ni ϕ i y + λ ∑︀ j>k b nj ϕ j y for every n > k, we readily see that ϕ k y = ∑︁ 0≤i<k c m+1,i ϕ i y + λ m+1 ∑︁ n>k d m+1,n ϕ n y, for suitable c m+1,i , d m+1,n ∈ R, almost all zero.
Since R is archimedean, there exists t > 0 such that λ t y = 0. Under this circumstance, we get λ t ∑︀ n>k d tn ϕ n y = ∑︀ n>k d tn ϕ n (λ t y) = 0, whence ϕ k y = ∑︀ 0≤i<k c t,i ϕ i y, as required.
Proposition 3.4. Let G be a torsion R-module, ϕ ∈ End R (G), x ∈ G, and assume that T(ϕ, x) is not finitely generated. Then T k is a pure submodule of Tn, for any 0 ≤ k < n. In particular, T 0 = 0 < T 1 < · · · < T n−1 < Tn is a pure-composition series of Tn, with increasing annihilator sequence.
Proof. We make induction on k. Let us prove that T 1 = Rx is pure in Tn = ⟨x, ϕx, . . . , ϕ n−1 x⟩. We assume, for a contradiction, that Rx is not pure in Tn. Then there exists a relation 0 ≠ ax = r ∑︀ j≥1 a j ϕ j x (a, a i , r ∈ R), where ax ∉ rRx, i.e., r = aλ for some λ ∈ P. We set y = ax. Then we get y = λ ∑︀ j>1 a i ϕ j y, so y = 0 by Lemma 4.2, a contradiction.
Let the statement be true for k − 1, and assume, for a contradiction, that T k is not pure in Tn. Then there exists a relation In conclusion, we may assume that c i = b i /b k ∈ R for every i < k, and ra j /b k = λd j , where d j ∈ R and λ ∈ P, for every j > k.
Let us set y = b k x. The above discussion shows that hence ϕ k y ∈ ⟨y, ϕy, . . . , ϕ k−1 y⟩ ≤ T k−1 , in view of Lemma 4.3. But ϕ k y = b k ϕ k x ∉ T k−1 , a contradiction. We conclude that T k is necessarily pure in Tn.
In particular, it follows that T 0 = 0 < T 1 < · · · < T n−1 < Tn is a pure-composition series of Tn. It remains to show that this pure-composition series has increasing annihilator sequence. For 0 < k ≤ n, let A k = Ann(T k /T k−1 ). Pick r ∈ A k ; then, from rϕ k+1 x = ϕ(rϕ k x) ∈ ϕT k−1 ≤ T k , we get r ∈ A k+1 . We conclude that A k ⊆ A k+1 and therefore A 1 = Ann(x) ⊆ A 2 ⊆ · · · ⊆ An.
We are in the position to compute the v-entropy of ϕ restricted to the cyclic ϕ-trajectory generated by x ∈ G. Naturally, we consider only the non-trivial case where T(ϕ, x) is not finitely generated.

Theorem 3.5. Let G be a torsion R-module, ϕ ∈ End
Proof. To simplify the symbols, in the proof we will denote ϕ| T by ϕ. Since every finitely generated submodule of T is contained in some T j , we have entv(ϕ) = sup j {H(ϕ, T j )}

Note that Tn(ϕ, T j ) = T n+j−1 , and recall that v(A 1 ) ≥ v(A 2 ) ≥ . . . and v(A) = inf k v(A k ). Now take any integer k > 0. If n + j − 1 > k, by Proposition 4.4 and (2), we get
Since j and k were arbitrary, we get

Let us verify the reverse inequality entv(ϕ) ≥ v(A). In fact, Lv
A result less precise but more general than Theorem 4.5 has some interest.

Proposition 3.6. Let ϕ be an endomorphism of the torsion R-module G such that G = T(ϕ, F) for a suitable F ∈ F(G). Then ent(ϕ) ≤ Lv(F). In particular, ent(ϕ) < ∞.
Proof. Since G = T(ϕ, F), every finitely generated submodule of G is contained in some T j = T j (ϕ, F).
In our final result we get a special case of the Addition Theorem, namely, when the R-module G coincides with the cyclic trajectory T(ϕ, x), for some ϕ ∈ End R (G), x ∈ G, and the ϕ-invariant submodule coincides with aG, a ∈ P. We use a direct argument, based on Theorem 4.5, that is much simpler and shorter than the discussion leading to the proof of the general result in [8].

Proposition 3.7. Let G be a torsion R-module, not finitely generated, such that G = T(ϕ, x) for suitable ϕ ∈
End R (G) and x ∈ G. Take any a ∈ P, and consider the fully invariant submodule aG = T(ϕ, ax) and the induced mapφ : G/aG → G/aG. Then we have Proof. For n > 0, let Tn = ⟨x, ϕx, . . . , ϕ n−1 x⟩ and An = Ann(Tn /T n−1 ). We have seen in Theorem 4.5 that We firstly assume that aG is finitely generated. Then entv(ϕ| aG ) = 0 and there exists k > 0 such that T k ≥ aG. Consider the partialφ-trajectory Xn = Tn(φ, T k /aG) ≤ G/aG. Since T k ≥ aG, it follows that T j /T k ∼ = (T j /aG)/(T k /aG) for every j > k, and therefore the last terms of the increasing annihilator sequence of Xn are A k+1 ⊆ A k+2 ⊆ · · · ⊆ A k+n . Under the present circumstances, an argument similar to the proof of Theorem 4.5 shows that entv(φ) = v(A), hence ( †) holds in this case. Now we assume that aG = T(ϕ, ax) is not finitely generated; equivalently, gen(aT j ) = j for every j > 0. Note that aG not finitely generated implies that a ∉ A j , i.e., aR ⊃ A j , for every j > 0. Since Tn is pure in T n+1 , a simple computation shows that aT j has annihilator sequence a −1 A 1 ⊆ a −1 A 2 ⊆ · · · ⊆ a −1 A j . Then Theorem 4.5, applied to y = ax yields entv(ϕ| aG ) = v(a −1 A) = v(A) − v(a). Thus, in order to verify ( †), it remains to show that entv(φ) = v(a). Let us verify that Assume, for a contradiction, that the sum is not direct. Then there exists k > 0 such that where b k ∉ aR. If m > k, we get acm ∈ Am, hence acm ϕ m x = acm ∑︀ m−1 i=0 d i ϕ i x, since T m−1 is pure in Tm. Thus we get a shorter relation where b k − acm d k ∉ aR. Repeating the procedure, after a finite number of steps we get an equality of the form where e k ∉ aR. But this means that e k ∈ A k ⊂ aR, impossible. We reached a contradiction; then, necessarily, the sum is direct.
The desired conclusion follows, applying Theorem 4.5 to G/aG andφ.

Computation of v-entropies.
The aim of this section is to get a good description of the cyclic ϕ-trajectories contained in the torsion Rmodule G, where ϕ ∈ End R (G).
To simplify the notation, in what follows, for assigned ϕ ∈ End R (G) and x ∈ G, we will denote by Tn the n-partial trajectory Tn(ϕ, x). Proposition 4.1. Let G be a torsion R-module, ϕ ∈ End R (G), x ∈ G. If the ϕ-trajectory T(ϕ, x) is not finitely generated, then gen Tn = gen⟨x, ϕx, . . . , ϕ n−1 x⟩ = n for every n > 0.
Proof. Let us assume, for a contradiction, that gen Tm < m for some m > 0. Then ∑︀ m−1 i=0 a i ϕ i x = 0, for suitable a i ∈ R not all in P. From this relation it follows that there exist polynomials p 0 , p 1 , with p 0 monic, such that p 0 (ϕ)x = λp 1 (ϕ)x, where λ ∈ P. Now take k > 0 large enough such that λ k x = 0. This is possible, since R is archimedean. Then we get p k 0 (ϕ)x = p k 1 (ϕ)(λ k x) = 0. We conclude that x is annihilated by p k 0 (ϕ), where p k 0 is a monic polynomial, of degree h, say, and so Tn ⊆ T h+1 for every n > 0. It follows that T(ϕ, x) = T h+1 is finitely generated, impossible.
The following two lemmas are crucial to get the next Proposition 4.4.

Lemma 4.2.
Let G be a torsion R-module, ϕ ∈ End R (G), y ∈ G. If y = λ ∑︀ m i=1 a i ϕ i y, for some m > 0, a i ∈ R and λ ∈ P, then y = 0.
Proof. We assume, for a contradiction, that y ≠ 0, hence 0 ≠ Ann y ≠ R. Since R is archimedean, there exists r ∉ Ann y such that λry = 0. Then we get We reached a contradiction.

Lemma 4.3.
Let G be a torsion R-module, ϕ ∈ End R (G), y ∈ G. If ϕ k y = ∑︀ 0≤i<k a i ϕ i y + λ ∑︀ j>k b j ϕ j y, for some a i , b j ∈ R (almost all zero) and λ ∈ P, then ϕ k y = ∑︀ 0≤i<k c i ϕ i y, for suitable c i ∈ R.
Proof. Firstly we prove that, for every n ≥ k, we can write a relation ϕ n y = ∑︁ 0≤i<k a ni ϕ i y + λ ∑︁ j>k b nj ϕ j y, for suitable a ni , b nj ∈ R, almost all zero. We make induction on n. The hypothesis yields the case n = k. Assume that the above relation is valid. Then we get for suitable a n+1,i , b n+1,j ∈ R, almost all zero. Now we prove that, for every m > 0, we can write a relation for suitable c mi , dmn ∈ R, almost all zero. We make induction on m. The case m = 1 is covered by the hypothesis. Assume that the above relation is valid. Since ϕ n y = ∑︀ 0≤i<k a ni ϕ i y + λ ∑︀ j>k b nj ϕ j y for every n > k, we readily see that for suitable c m+1,i , d m+1,n ∈ R, almost all zero.
Since R is archimedean, there exists t > 0 such that λ t y = 0. Under this circumstance, we get λ t ∑︀ n>k d tn ϕ n y = ∑︀ n>k d tn ϕ n (λ t y) = 0, whence ϕ k y = ∑︀ 0≤i<k c t,i ϕ i y, as required.

Proposition 4.4.
Let G be a torsion R-module, ϕ ∈ End R (G), x ∈ G, and assume that T(ϕ, x) is not finitely generated. Then T k is a pure submodule of Tn, for any 0 ≤ k < n. In particular, T 0 = 0 < T 1 < · · · < T n−1 < Tn is a pure-composition series of Tn, with increasing annihilator sequence.
Proof. We make induction on k. Let us prove that T 1 = Rx is pure in Tn = ⟨x, ϕx, . . . , ϕ n−1 x⟩. We assume, for a contradiction, that Rx is not pure in Tn. Then there exists a relation 0 ≠ ax = r ∑︀ j≥1 a j ϕ j x (a, a i , r ∈ R), where ax ∉ rRx, i.e., r = aλ for some λ ∈ P. We set y = ax. Then we get y = λ ∑︀ j>1 a i ϕ j y, so y = 0 by Lemma 4.2, a contradiction.
Let the statement be true for k − 1, and assume, for a contradiction, that T k is not pure in Tn. Then there exists a relation In conclusion, we may assume that c i = b i /b k ∈ R for every i < k, and ra j /b k = λd j , where d j ∈ R and λ ∈ P, for every j > k.
Let us set y = b k x. The above discussion shows that hence ϕ k y ∈ ⟨y, ϕy, . . . , ϕ k−1 y⟩ ≤ T k−1 , in view of Lemma 4.3. But ϕ k y = b k ϕ k x ∉ T k−1 , a contradiction. We conclude that T k is necessarily pure in Tn.
In particular, it follows that T 0 = 0 < T 1 < · · · < T n−1 < Tn is a pure-composition series of Tn. It remains to show that this pure-composition series has increasing annihilator sequence. For 0 < k ≤ n, let A k = Ann(T k /T k−1 ). Pick r ∈ A k ; then, from rϕ k+1 x = ϕ(rϕ k x) ∈ ϕT k−1 ≤ T k , we get r ∈ A k+1 . We conclude that A k ⊆ A k+1 and therefore A 1 = Ann(x) ⊆ A 2 ⊆ · · · ⊆ An.
We are in the position to compute the v-entropy of ϕ restricted to the cyclic ϕ-trajectory generated by x ∈ G. Naturally, we consider only the non-trivial case where T(ϕ, x) is not finitely generated.
Proof. To simplify the symbols, in the proof we will denote ϕ| T by ϕ. Since every finitely generated submodule of T is contained in some T j , we have . Now take any integer k > 0. If n + j − 1 > k, by Proposition 4.4 and (2), we get Since j and k were arbitrary, we get Let us verify the reverse inequality entv(ϕ) ≥ v(A). In fact, Lv(T n+j−1 ) ≥ (n + j − 1)v(A), whence A result less precise but more general than Theorem 4.5 has some interest. The desired conclusion follows, since j > 0 was arbitrary.
In our final result we get a special case of the Addition Theorem, namely, when the R-module G coincides with the cyclic trajectory T(ϕ, x), for some ϕ ∈ End R (G), x ∈ G, and the ϕ-invariant submodule coincides with aG, a ∈ P. We use a direct argument, based on Theorem 4.5, that is much simpler and shorter than the discussion leading to the proof of the general result in [8].
Proposition 4.7. Let G be a torsion R-module, not finitely generated, such that G = T(ϕ, x) for suitable ϕ ∈ End R (G) and x ∈ G. Take any a ∈ P, and consider the fully invariant submodule aG = T(ϕ, ax) and the induced mapφ : G/aG → G/aG. Then we have entv(ϕ) = entv(ϕ| aG ) + entv(φ) ( †) Proof. For n > 0, let Tn = ⟨x, ϕx, . . . , ϕ n−1 x⟩ and An = Ann(Tn /T n−1 ). We have seen in Theorem 4.5 that entv(ϕ) = v(A), where A = ⋃︀ j>0 A j . We firstly assume that aG is finitely generated. Then entv(ϕ| aG ) = 0 and there exists k > 0 such that T k ≥ aG. Consider the partialφ-trajectory Xn = Tn(φ, T k /aG) ≤ G/aG. Since T k ≥ aG, it follows that T j /T k ∼ = (T j /aG)/(T k /aG) for every j > k, and therefore the last terms of the increasing annihilator sequence of Xn are A k+1 ⊆ A k+2 ⊆ · · · ⊆ A k+n . Under the present circumstances, an argument similar to the proof of Theorem 4.5 shows that entv(φ) = v(A), hence ( †) holds in this case. Now we assume that aG = T(ϕ, ax) is not finitely generated; equivalently, gen(aT j ) = j for every j > 0. Note that aG not finitely generated implies that a ∉ A j , i.e., aR ⊃ A j , for every j > 0. Since Tn is pure in T n+1 , a simple computation shows that aT j has annihilator sequence a −1 A 1 ⊆ a −1 A 2 ⊆ · · · ⊆ a −1 A j . Then where e k ∉ aR. But this means that e k ∈ A k ⊂ aR, impossible. We reached a contradiction; then, necessarily, the sum is direct.
The desired conclusion follows, applying Theorem 4.5 to G/aG andφ.