Functional identities on upper triangular matrix rings

He Yuan; Liangyun Chen

doi:10.1515/math-2020-0018

Article Open Access

Functional identities on upper triangular matrix rings

He Yuan and Liangyun Chen

Published/Copyright: March 20, 2020

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$Open Mathematics$

From the journal Open Mathematics Volume 18 Issue 1

Abstract

Let R be a subset of a unital ring Q such that 0 ∈ R. Let us fix an element t ∈ Q. If R is a (t; d)-free subset of Q, then T_n(R) is a (t′; d)-free subset of T_n(Q), where t′ ∈ T_n(Q), tll′ = t, l = 1, 2, …, n, for any n ∈ N.

Keywords: functional identity; (t; d)-free subset; upper triangular matrix ring

MSC 2010: 16R60; 16S50

1 Introduction

A functional identity (FI) is a kind of identity relation involving the elements in a ring and the functions; more precisely, the values of the functions are multiplied by the elements. The theory of functional identity can be used to describe the representation of the functions in the identity or the structure of the ring involved. The theory can be used to solve various problems in different fields.

FI theory is a relatively new one. Its root is the Ph.D. Thesis of Brešar in 1990, and then he published a series of papers on functional identities, among which the form of commuting maps on a ring is considered to be the first application of the theory [1]. Beidar [2] established the theory on the basis of some results in 1998. In 2000, the theory culminated in the works by Beidar and Chebotar on d-free sets [3, 4]. In the past few years, the focus has been on the applications of the theory. By using the theory, the conjectures of Lie homomorphisms and Lie derivations in associative rings proposed by Herstein are solved [5, 6, 7]. In fact, the main motivation of studying the theory is to solve the conjectures. For a full account on the theory of functional identities, the reader is referred to the book [8].

Cheung [9] described the form of commuting linear maps for a certain class of triangular algebras in 2001. Later, several problems on certain types of maps on triangular rings and algebras have been studied. In 2013, Eremita [10] initiated the study of functional identities on triangular rings. Further results have been obtained in [11, 12]. In 2016, Eremita [13] studied d-free subset of upper triangular matrix rings. Later, Wang [14] gave an improvement of a result of Eremita [13], which also presented

Theorem 1.1

[14, Theorem 1.2] Let R be a subset of a unital ring Q such that 0 ∈ R. If R is a d-free subset of Q, then T_n(R) is a d-free subset of T_n(Q) for any n ∈ N.

The aim of the paper is to give an improvement of Theorem 1.1, that is, to study (t; d)-free subset of upper triangular matrix rings.

Theorem 1.2

Let R be a subset of a unital ring Q such that 0 ∈ R. Let us fix an element t ∈ Q. If R is a (t; d)-free subset of Q, then T_n(R) is a (t′; d)-free subset of T_n(Q), where t′ ∈ T_n(Q), tll′ = t, l = 1, 2, …, n, for any n ∈ N.

2 Preliminaries

Let R be a nonempty subset of an associative unital ring Q with center Z(Q). By T_n(R) we denote the set of all n × n upper triangular matrices over R, and by T_n(Q) the upper triangular matrix ring over Q. Suppose that m ∈ ℕ, the set of all positive integers, 𝓘, 𝓙 ⊆ {1, 2, …, m}. Let a, b be nonnegative integers, and E_iu : R^m−1 → Q, i ∈ 𝓘, 0 ⩽ u ⩽ a, and F_jv : R^m−1 → Q, j ∈ 𝓙, 0 ⩽ v ⩽ b, be arbitrary maps. For a fixed t ∈ Q we consider identities

∑i∈I∑u=0aEiu(x¯mi)xitu+∑j∈J∑v=0btvxjFjv(x¯mj)=0, (2.1)

∑i∈I∑u=0aEiu(x¯mi)xitu+∑j∈J∑v=0btvxjFjv(x¯mj)∈Z(Q) (2.2)

for all x_m ∈ R^m, where x_m = (x₁,x₂, …, x_m) and x¯mi = (x₁, …, x_i−1, x_i+1, …, x_m). A natural possibility when (2.1) (and hence also (2.2)) is fulfilled is when there exist maps

piujv:Rm−2→Q,i∈I,j∈J,i≠j,0⩽u⩽a,0⩽v⩽b,λkuv:Rm−1→Z(Q),k∈I∪J,0⩽u⩽a,0⩽v⩽b,

such that

Eiu(x¯mi)=∑j∈Jj≠i∑v=0btvxjpiujv(x¯mij)+∑v=0bλiuv(x¯mi)tv,Fjv(x¯mj)=−∑i∈Ii≠j∑u=0apiujv(x¯mij)xitu−∑u=0aλjuv(x¯mj)tu,λkuv=0ifk∉I∩J, (2.3)

where x¯mij=x¯mji = (x₁, …, x_i−1, x_i+1, …, x_j−1, x_j+1, …, x_m). One can easily verify that (2.3) is indeed a solution of both (2.1) and (2.2). We say that (2.3) is a standard solution of (2.1) and (2.2).

Definition 2.1

[8, Definition 3.8] R is said to be a (t; d)-free subset of Q, where t ∈ Q and d ∈ ℕ, if for all 𝓘, 𝓙 ⊆ {1, 2, …, m}, and all integers a, b ⩾ 0 the following two conditions are satisfied:

If max{∣𝓘∣ + a, ∣𝓙∣ + b} ⩽ d, then (2.1) implies (2.3).
If max{∣𝓘∣ + a, ∣𝓙∣ + b} ⩽ d − 1, then (2.2) implies (2.3).

The case when one of the subsets 𝓘, 𝓙 is empty is also included. Namely, according to the usual convention, that the sum over the empty set is zero, (2.1) can be rewritten as

∑i∈I∑u=0aEiu(x¯mi)xitu=0for all x¯m∈Rm (2.4)

if 𝓙 = ∅ or as

∑j∈J∑v=0btvxjFjv(x¯mj)=0for all x¯m∈Rm (2.5)

if 𝓘 = ∅. Similarly, (2.2) reduces to

∑i∈I∑u=0aEiu(x¯mi)xitu∈C(t)for all x¯m∈Rm (2.6)

if 𝓙 = ∅ or to

∑j∈J∑v=0btvxjFjv(x¯mj)∈C(t)for all x¯m∈Rm (2.7)

if 𝓘 = ∅, where C(t) denotes the centralizer of t in Q. According to [8, Lemma 3.9], we have

Lemma 2.2

Let R be a (t; d)-free subset of Q. Then:

If ∣𝓘∣ + a ⩽ d, then (2.4) implies that each E_iu = 0.
If ∣𝓙∣ + b ⩽ d, then (2.5) implies that each F_jv = 0.
If ∣𝓘∣ + a ⩽ d − 1, then (2.6) implies that each E_iu = 0.
If ∣𝓙∣ + b ⩽ d − 1, then (2.7) implies that each F_jv = 0.

Set

Nm={1,2,…,m}andNn⊲={(i,j)∈Nn×Nn|i⩽j}.

Suppose that A = (a_ij) ∈ M_m×n(R). Set (A)_ij = a_ij and

As⌜=a11a12⋯a1sa21a22⋯a2s⋮⋮⋱⋮as1as2⋯assfor1⩽s⩽min{m,n},As,t⌝=a1ta2t⋮astfor1⩽s⩽m,1⩽t⩽n,A⌟=amn.

Suppose that A = (a_ij) ∈ T_n(R). For brevity, we set

A⌜=An−1⌜,A⌝=An−1,n⌝andA⌟=ann.

Next, let S be a set, for each map E : S → T_n(Q) the maps

E⌜:S→Tn−1(Q),E⌝:S→M(n−1)×1(Q),E⌟:S→Q

are defined by

E⌜(s)=E(s)⌜,E⌝(s)=E(s)⌝,E⌟(s)=E(s)⌟.

3 The main result

We are now ready to prove our main theorem.

Proof

Proof of Theorem 1.2 Suppose that R is a (t; d)-free subset of Q. In order to prove that T_n(R) is a (t′; d)-free subset of T_n(Q) for each n ∈ N, where t′ ∈ T_n(Q), tll′ = t, l = 1, 2, …, n, we proceed by induction on n. If n = 1 the result follows trivially. Now, suppose that n ⩾ 2. Let a, b be nonnegative integers. Let m ∈ ℕ and let I, J ⊆ {1, 2, …, m}, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b. Further, let E_iu, F_jv : T_n(R)^m−1 → T_n(Q) be arbitrary maps. According to our notation introduced we have

Eiu(x¯mi)=Eiu⌜(x¯mi)Eiu⌝(x¯mi)0Eiu⌟(x¯mi),Fjv(x¯mj)=Fjv⌜(x¯mj)Fjv⌝(x¯mj)0Fjv⌟(x¯mj),xi=xi⌜xi⌝0xi⌟,xj=xj⌜xj⌝0xj⌟,(t′)u=(t′⌜)utu′¯0(t′⌟)uand(t′)v=(t′⌜)vtv′¯0(t′⌟)v. (3.1)

Our first aim is to prove that the functional identity

∑i∈I∑u=0aEiu(x¯mi)xi(t′)u+∑j∈J∑v=0b(t′)vxjFjv(x¯mj)=0for allx¯m∈Tn(R)m (3.2)

has only a standard solution, if max{∣𝓘∣ + a, ∣𝓙∣ + b} ⩽ d. Suppose that max{∣𝓘∣ + a, ∣𝓙∣ + b} ⩽ d. Note that (3.2) yields the following identities

∑i∈I∑u=0aEiu⌜(x¯mi)xi⌜(t′⌜)u+∑j∈J∑v=0b(t′⌜)vxj⌜Fjv⌜(x¯mj)=0, (3.3)

∑i∈I∑u=0aEiu⌟(x¯mi)xi⌟(t′⌟)u+∑j∈J∑v=0b(t′⌟)vxj⌟Fjv⌟(x¯mj)=0, (3.4)

∑i∈I∑u=0aEiu⌜(x¯mi)xi⌜tu′¯+∑i∈I∑u=0aEiu⌜(x¯mi)xi⌝(t′⌟)u+∑i∈I∑u=0aEiu⌝(x¯mi)xi⌟(t′⌟)u+∑j∈J∑v=0b(t′⌜)vxj⌜Fjv⌝(x¯mj)+∑j∈J∑v=0b(t′⌜)vxj⌝Fjv⌟(x¯mj)+∑j∈J∑v=0btv′¯xj⌟Fjv⌟(x¯mj)=0 (3.5)

for all x_m ∈ T_n(R)^m. First, we consider (3.3). For arbitrary matrices x₁, x₂, …, x_m ∈ T_n(R), we fix xi⌝,xi⌟ and set y_i = xi⌜ for all i. Define maps E_iu, F_jv:T_n−1(R)^m−1 → T_n−1(Q), i ∈ 𝓘, j ∈ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, by

E¯iu(y¯mi)=Eiu⌜(x¯mi)andF¯jv(y¯mj)=Fjv⌜(x¯mj) (3.6)

for all y_m ∈ T_n−1(R)^m. Thus, (3.3) can be rewritten as

∑i∈I∑u=0aE¯iu(y¯mi)yi(t′⌜)u+∑j∈J∑v=0b(t′⌜)vyjF¯jv(y¯mj)=0 (3.7)

for all y_m ∈ T_n−1(R)^m. According to the induction hypothesis T_n−1(R) is a (t′^⌜; d)-free subset of T_n−1(Q), where t′^⌜ ∈ T_n−1(Q), and so there exist maps

p¯iujv:Tn−1(R)m−2→Tn−1(Q),i∈I,j∈J,i≠j,0⩽u⩽a,0⩽v⩽b,λ¯kuv:Tn−1(R)m−1→Z(Q),k∈I∪J,0⩽u⩽a,0⩽v⩽b,

such that

E¯iu(y¯mi)=∑j∈Jj≠i∑v=0b(t′⌜)vyjp¯iujv(y¯mij)+∑v=0bλ¯iuv(y¯mi)(t′⌜)v,F¯jv(y¯mj)=−∑i∈Ii≠j∑u=0ap¯iujv(y¯mij)yi(t′⌜)u−∑u=0aλ¯juv(y¯mj)(t′⌜)u,λ¯kuv=0ifk∉I∩J (3.8)

for all y_m ∈ T_n−1(R)^m.

We claim that (x₁, …, x_m) → p_iujv and (x₁, …, x_m) → λ_kuv are well-defined maps. Suppose that there exist maps p′_iujv and λ′_kuv such that (3.8), then

E¯iu(y¯mi)=∑j∈Jj≠i∑v=0b(t′⌜)vyjp¯iujv(y¯mij)+∑v=0bλ¯iuv(y¯mi)(t′⌜)v=∑j∈Jj≠i∑v=0b(t′⌜)vyjp′¯iujv(y¯mij)+∑v=0bλ′¯iuv(y¯mi)(t′⌜)v

for all y_m ∈ T_n−1(R)^m. Hence

∑j∈Jj≠i∑v=0b(t′⌜)vyj(p¯iujv(y¯mij)−p′¯iujv(y¯mij))=∑v=0b(λ′¯iuv(y¯mi)−λ¯iuv(y¯mi))(t′⌜)v∈C(t′⌜)

for all y_m ∈ T_n−1(R)^m. Since T_n−1(R) is a (t′^⌜; d)-free subset of T_n−1(Q) it follows that p_iujv = p′_iujv and λ_kuv = λ′_kuv, where i ∈ 𝓘, j ∈ 𝓙, i ≠ j, k ∈ 𝓘 ∪ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b.

According to (3.8), we set

(x1,…,xi,…,xj,…,xm)→p¯iujv;(x1,…,0,…,xj,…,xm)→g¯iujv;(x1,…,0,…,0,…,xm)→h¯iujv;(x1,…,xk,…,xm)→λ¯kuv;(x1,…,0,…,xm)→η¯kuv,

where i ∈ 𝓘, j ∈ 𝓙, i ≠ j, k ∈ 𝓘 ∪ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b. Next, we claim that

p¯iujv=h¯iujvandλ¯kuv=η¯kuv.

In fact, it follows from (3.6) and (3.8) that

E¯iu(y¯mi)=∑j∈Jj≠i∑v=0b(t′⌜)vyjp¯iujv(y¯mij)+∑v=0bλ¯iuv(y¯mi)(t′⌜)v=∑j∈Jj≠i∑v=0b(t′⌜)vyjg¯iujv(y¯mij)+∑v=0bη¯iuv(y¯mi)(t′⌜)v

for all y_m ∈ T_n−1(R)^m. Thus, p_iujv = g_iujv and λ_iuv = η_iuv. Similarly, we have g_iujv = h_iujv and λ_juv = η_juv. It follows that p_iujv = h_iujv and λ_kuv = η_kuv, i ∈ 𝓘, j ∈ 𝓙, i ≠ j, k ∈ 𝓘 ∪ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b. That is, p_iujv is independent of x_i and x_j, and λ_kuv is independent of x_k.

We now define maps p_iujv : T_n(R)^m−2 → T_n−1(Q) and λ_kuv : T_n(R)^m−1 → Z(Q), i ∈ 𝓘, j ∈ 𝓙, i ≠ j, k ∈ 𝓘 ∪ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, by

piujv(x¯mij)=p¯iujv(y¯mij)andλkuv(x¯mk)=λ¯kuv(y¯mk).

Then (3.8) implies that

Eiu⌜(x¯mi)=∑j∈Jj≠i∑v=0b(t′⌜)vxj⌜piujv(x¯mij)+∑v=0bλiuv(x¯mi)(t′⌜)v,Fjv⌜(x¯mj)=−∑i∈Ii≠j∑u=0apiujv(x¯mij)xi⌜(t′⌜)u−∑u=0aλjuv(x¯mj)(t′⌜)u,λkuv=0ifk∉I∩J (3.9)

for all x_m ∈ T_n(R)^m.

In a similar manner, one can prove that (3.4) implies the existence of maps q_iujv : T_n(R)^m−2 → Q, and μ_kuv : T_n(R)^m−1 → Z(Q), i ∈ 𝓘, j ∈ 𝓙, i ≠ j, k ∈ 𝓘 ∪ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, such that

Eiu⌟(x¯mi)=∑j∈Jj≠i∑v=0b(t′⌟)vxj⌟qiujv(x¯mij)+∑v=0bμiuv(x¯mi)(t′⌟)v,Fjv⌟(x¯mj)=−∑i∈Ii≠j∑u=0aqiujv(x¯mij)xi⌟(t′⌟)u−∑u=0aμjuv(x¯mj)(t′⌟)u,μkuv=0ifk∉I∩J (3.10)

for all x_m ∈ T_n(R)^m.

Next, we consider (3.5). Using (3.9) and (3.10) we can rewrite (3.5) as

∑i∈I∑j∈Jj≠i∑u=0a∑v=0b(t′⌜)vxj⌜piujv(x¯mij)xi⌜tu′¯+∑i∈I∑u=0a∑v=0bλiuv(x¯mi)(t′⌜)vxi⌜tu′¯+∑i∈I∑j∈Jj≠i∑u=0a∑v=0b(t′⌜)vxj⌜piujv(x¯mij)xi⌝(t′⌟)u+∑i∈I∑u=0a∑v=0bλiuv(x¯mi)(t′⌜)vxi⌝(t′⌟)u+∑i∈I∑u=0aEiu⌝(x¯mi)xi⌟(t′⌟)u+∑j∈J∑v=0b(t′⌜)vxj⌜Fjv⌝(x¯mj)−∑j∈J∑i∈Ii≠j∑v=0b∑u=0a(t′⌜)vxj⌝qiujv(x¯mij)xi⌟(t′⌟)u−∑j∈J∑v=0b∑u=0a(t′⌜)vxj⌝μjuv(x¯mj)(t′⌟)u−∑j∈J∑i∈Ii≠j∑v=0b∑u=0atv′¯xj⌟qiujv(x¯mij)xi⌟(t′⌟)u−∑j∈J∑v=0b∑u=0atv′¯xj⌟μjuv(x¯mj)(t′⌟)u=0

for all x_m ∈ T_n(R)^m. That is,

∑i∈I∑u=0a(Eiu⌝(x¯mi)−∑j∈Jj≠i∑v=0b(t′⌜)vxj⌝qiujv(x¯mij)−∑j∈Jj≠i∑v=0btv′¯xj⌟qiujv(x¯mij)−∑v=0btv′¯μiuv′(x¯mi))xi⌟(t′⌟)u+∑j∈J∑v=0b(t′⌜)vxj⌜(Fjv⌝(x¯mj)+∑i∈Ii≠j∑u=0apiujv(x¯mij)xi⌜tu′¯+∑i∈Ii≠j∑u=0apiujv(x¯mij)xi⌝(t′⌟)u+∑u=0aλjuv′(x¯mj)tu′¯)+∑k∈I∩J∑u=0a∑v=0b(λkuv(x¯mk)−μkuv(x¯mk))(t′⌜)vxk⌝(t′⌟)u=0 (3.11)

for all x_m ∈ T_n(R)^m, where

μiuv′=0i∉I∩J,μiuvi∈I∩J,andλjuv′=0j∉I∩J,λjuvj∈I∩J.

Now we define maps G_iu, H_jv : T_n(R)^m−1 → M_(n−1)×1(Q), α_kuv : T_n(R)^m−1 → Z(Q), where i ∈ 𝓘, j ∈ 𝓙, i ≠ j, k ∈ 𝓘 ∩ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, by

Giu(x¯mi)=Eiu⌝(x¯mi)−∑j∈Jj≠i∑v=0b(t′⌜)vxj⌝qiujv(x¯mij)−∑j∈Jj≠i∑v=0btv′¯xj⌟qiujv(x¯mij)−∑v=0btv′¯μiuv′(x¯mi),Hjv(x¯mj)=Fjv⌝(x¯mj)+∑i∈Ii≠j∑u=0apiujv(x¯mij)xi⌜tu′¯+∑i∈Ii≠j∑u=0apiujv(x¯mij)xi⌝(t′⌟)u+∑u=0aλjuv′(x¯mj)tu′¯,αkuv(x¯mk)=λkuv(x¯mk)−μkuv(x¯mk) (3.12)

for all x_m ∈ T_n(R)^m. Thus, (3.11) can be rewritten as

∑i∈I∑u=0aGiu(x¯mi)xi⌟tu+∑j∈J∑v=0b(t′⌜)vxj⌜Hjv(x¯mj)+∑k∈I∩J∑u=0a∑v=0bαkuv(x¯mk)(t′⌜)vxk⌝tu=0 (3.13)

for all x_m ∈ T_n(R)^m, where t′ ∈ T_n(Q), tll′ = t, l = 1, 2, …, n.

Now, we claim that α_kuv = 0 for all k ∈ 𝓘 ∩ 𝓙 and there exist maps r_iujv : T_n(R)^m−2 → M_(n−1)×1(Q), i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, such that

Giu(x¯mi)=∑j∈Jj≠i∑v=0b(t′⌜)vxj⌜riujv(x¯mij),Hjv(x¯mj)=−∑i∈Ii≠j∑u=0ariujv(x¯mij)xi⌟tu

for all x_m ∈ T_n(R)^m. We proceed by induction on n. If n = 2, then we can rewrite (3.13) as

∑i∈I∑u=0aGiu(x¯mi)(xi)22tu+∑j∈J∑v=0btv(xj)11Hjv(x¯mj)+∑k∈I∩J∑u=0a∑v=0bαkuv(x¯mk)tv(xk)12tu=0 (3.14)

for all x_m ∈ T₂(R)^m. For arbitrary matrices x₁, x₂, …, x_m ∈ T₂(R), we set

yi=(xi)22,yi+m=(xi)11,yi+2m=(xi)12

for all i. We can define maps G_iu, H_(j+m)v : R^3m−1 → Q and α_(k+2m)uv : R^3m−1 → Z(Q), i ∈ 𝓘, j ∈ 𝓙, k ∈ 𝓘 ∩ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, by

G¯iu(y¯3mi)=Giu(x¯mi),H¯(j+m)v(y¯3mj+m)=Hjv(x¯mj),α¯(k+2m)uv(y¯3mk+2m)=αkuv(x¯mk) (3.15)

for all y_3m ∈ R^3m. Then (3.14) can be rewritten as

∑i∈I∑u=0aG¯iu(y¯3mi)yitu+∑j∈J∑v=0btvyj+mH¯(j+m)v(y¯3mj+m)+∑k∈I∩J∑u=0a∑v=0bα¯(k+2m)uv(y¯3mk+2m)tvyk+2mtu=0 (3.16)

for all y_3m ∈ R^3m. We claim that α_(k+2m)uv = 0 for all k ∈ 𝓘 ∩ 𝓙. We proceed by induction on ∣𝓘 ∩ 𝓙∣.

Suppose that ∣𝓘 ∩ 𝓙∣ = 1. Let 𝓘 ∩ 𝓙 = {k₀}. Taking y_k₀ = 0 in (3.16) we obtain

∑i∈I∖{k0}∑u=0aG¯iu(y¯3mi)yitu+∑u=0a∑v=0bα¯(k0+2m)uv(y¯3mk0+2m)tvyk0+2mtu+∑j∈J∑v=0btvyj+mH¯(j+m)v(y¯3mj+m)=0 (3.17)

for all y_3m ∈ R^3m with y_k₀ = 0. Note that y_k₀, y_k₀+m do not appear in α_(k₀+2m)uv (y¯3mk0+2m) Since R is a (t; d)-free subset of Q and ∣𝓘 ∖ {k₀}∣ + 1 = ∣𝓘∣ it follows that there exist maps f_{(k₀+2m)u(j+m)v} : R^3m−2 → Q, j ∈ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, such that

∑v=0bα¯(k0+2m)uv(y¯3mk0+2m)tv=∑j∈J∖{k0}∑v=0btvyj+mf¯(k0+2m)u(j+m)v(y¯3m(k0+2m)(j+m))∈C(t)

for all y_3m ∈ R^3m. Since R is a (t; d)-free subset of Q and ∣𝓙 ∖ {k₀}∣ + b ⩽ d − 1 it follows from Lemma 2.2 that

f¯(k0+2m)u(j+m)v(y¯3m(k0+2m)(j+m))=0.

It is easy to check that α_(k₀+2m)uv (y¯3mk0+2m) = 0.

Suppose next that ∣𝓘 ∩ 𝓙∣ > 1. For any k₀ ∈ 𝓘 ∩ 𝓙, setting y_k₀ = 0 in (3.16) we get

∑i∈I∖{k0}∑u=0aG¯iu(y¯3mi)yitu+∑u=0a∑v=0bα¯(k0+2m)uv(y¯3mk0+2m)tvyk0+2mtu+∑j∈J∑v=0btvyj+mH¯(j+m)v(y¯3mj+m)+∑k∈(I∩J)∖{k0}∑u=0a∑v=0bα¯(k+2m)uv(y¯3mk+2m)tvyk+2mtu=0

for all y_3m ∈ R^3m with y_k₀ = 0. Since ∣(𝓘 ∩ 𝓙) ∖ {k₀}∣ = ∣𝓘 ∩ 𝓙∣ − 1 and max{∣𝓘 ∖ {k₀}∣ + 1 + a, ∣𝓙∣ + b} ⩽ d it follows from the induction hypothesis that α_(k+2m)uv (y¯3mk+2m) = 0 for all y_3m ∈ R^3m with y_k₀ = 0, where k ∈ (𝓘 ∩ 𝓙) ∖ {k₀}, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b. Thus

∑i∈I∖{k0}∑u=0aG¯iu(y¯3mi)yitu+∑u=0a∑v=0bα¯(k0+2m)uv(y¯3mk0+2m)tvyk0+2mtu+∑j∈J∑v=0btvyj+mH¯(j+m)v(y¯3mj+m)=0

for all y_3m ∈ R^3m with y_k₀ = 0. Using the same argument as (3.17) we can get that α_(k₀+2m)uv = 0. That is, α_(k+2m)uv = 0 for all k ∈ 𝓘 ∩ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b. Hence α_kuv = 0 for all k ∈ 𝓘 ∩ 𝓙, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b.

Next, we consider G_iu and H_(j+m)v. According to (3.16) we get

∑i∈I∑u=0aG¯iu(y¯3mi)yitu+∑j∈J∑v=0btvyj+mH¯(j+m)v(y¯3mj+m)=0 (3.18)

for all y_3m ∈ R^3m. Note that y_i+m does not appear in G_iu (y¯3mi) and y_j does not appear in H_(j+m)v (y¯3mj+m) Since R is a (t; d)-free subset of Q it follows that there exist maps f_iu(j+m)v:R^3m−2 → Q, i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, such that

G¯iu(y¯3mi)=∑j∈Jj≠i∑v=0btvyj+mf¯iu(j+m)v(y¯3mi(j+m)), (3.19)

H¯(j+m)v(y¯3mj+m)=−∑i∈Ii≠j∑u=0af¯iu(j+m)v(y¯3mi(j+m))yitu (3.20)

for all y_3m ∈ R^3m.

We claim that (x₁, …, x_m) → f_iu(j+m)v is a well-defined map. Namely, suppose that (x₁, …, x_m) → f′_iu(j+m)v, i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, is a map such that

G¯iu(y¯3mi)=∑j∈Jj≠i∑v=0btvyj+mf′¯iu(j+m)v(y¯3mi(j+m)),H¯(j+m)v(y¯3mj+m)=−∑i∈Ii≠j∑u=0af′¯iu(j+m)v(y¯3mi(j+m))yitu

for all y_3m ∈ R^3m. According to (3.19) we have

∑j∈Jj≠i∑v=0btvyj+m(f¯iu(j+m)v(y¯3mi(j+m))−f′¯iu(j+m)v(y¯3mi(j+m)))=0

for all y_3m ∈ R^3m. Since R is a (t; d)-free subset of Q it follows from Lemma 2.2 that f_iu(j+m)v = f′_iu(j+m)v for all i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b. Hence, (x₁, …, x_m) → f_iu(j+m)v is a well-defined map.

We now claim that f_iu(j+m)v is independent of x_i and x_j, where i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b. It follows from (3.15) and (3.19) that

∑j∈Jj≠i∑v=0btvyj+mf¯iu(j+m)v(x1,…,xi,…,xm)(y¯3mi(j+m))=∑j∈Jj≠i∑v=0btvyj+mf¯iu(j+m)v(x1,…,0,…,xm)(y¯3mi(j+m))

and then

∑j∈Jj≠i∑v=0btvyj+m(f¯iu(j+m)v(x1,…,xi,…,xm)(y¯3mi(j+m))−f¯iu(j+m)v(x1,…,0,…,xm)(y¯3mi(j+m)))=0

for all y_3m ∈ R^3m. Since R is a (t; d)-free subset of Q it follows from Lemma 2.2 that

f¯iu(j+m)v(x1,…,xi,…,xm)(y¯3mi(j+m))=f¯iu(j+m)v(x1,…,0,…,xm)(y¯3mi(j+m))

for all y_3m ∈ R^3m. This implies that f_iu(j+m)v is independent of x_i. Similarly, we can get from (3.15) and (3.20) that f_iu(j+m)v is independent of x_j as well.

Since each f_iu(j+m)v is determined uniquely and independent of x_i and x_j, we can define maps r_iujv : T₂(R)^m−2 → Q, i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, by

riujv(x¯mij)=f¯iu(j+m)v(y¯3mi(j+m))

for all x_m ∈ T₂(R)^m. We get from (3.19) and (3.20) that

Giu(x¯mi)=∑j∈Jj≠i∑v=0btv(xj)11riujv(x¯mij),Hjv(x¯mj)=−∑i∈Ii≠j∑u=0ariujv(x¯mij)(xi)22tu

for all x_m ∈ T₂(R)^m, where i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b.

Suppose that n ⩾ 3. We can define maps Giu⊤,Hjv⊤ : T_n(R)^m−1 → M_(n−2)×1(Q), Giu⊥,Hjv⊥ : T_n(R)^m−1 → Q, i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, by

Giu⊤(x¯mi)=(Giu(x¯mi))n−2,1⌝,Giu⊥(x¯mi)=Giu(x¯mi)⌟,Hjv⊤(x¯mi)=(Hjv(x¯mi))n−2,1⌝,Hjv⊥(x¯mi)=Hjv(x¯mi)⌟

for all x_m ∈ T_n(R)^m. Note that

Giu(x¯mi)=Giu⊤(x¯mi)Giu⊥(x¯mi),Hjv(x¯mj)=Hjv⊤(x¯mj)Hjv⊥(x¯mj),(xj)⌜=(xj)n−2⌜(xj)n−2,n−1⌝0(xj)n−1,n−1,(xk)⌝=(xk)n−2,n⌝(xk)n−1,n,(t′⌜)v=tn−2′⌜tn−2,n−1′⌝0tn−1,n−1′v=(tn−2′⌜)vtv′⌜¯0tv.

Then (3.13) yields the following identities

∑i∈I∑u=0aGiu⊤(x¯mi)xi⌟tu+∑j∈J∑v=0b(tn−2′⌜)v(xj)n−2⌜Hjv⊤(x¯mj)+∑j∈J∑v=0b(tn−2′⌜)v(xj)n−2,n−1⌝Hjv⊥(x¯mj)+∑j∈J∑v=0btv′⌜¯(xj)n−1,n−1Hjv⊥(x¯mj)+∑k∈I∩J∑u=0a∑v=0b(αkuv(x¯mk)(tn−2′⌜)v(xk)n−2,n⌝tu+αkuv(x¯mk)tv′⌜¯(xk)n−1,ntu)=0. (3.21)

and

∑i∈I∑u=0aGiu⊥(x¯mi)xi⌟tu+∑j∈J∑v=0btv(xj)n−1,n−1Hjv⊥(x¯mj)+∑k∈I∩J∑u=0a∑v=0bαkuv(x¯mk)tv(xk)n−1,ntu=0 (3.22)

for all x_m ∈ T_n(R)^m. Similar to the case of n = 2, we can get from (3.22) that α_kuv = 0 for all k ∈ 𝓘 ∩ 𝓙, and there exist maps q_iujv : T_n(R)^m−2 → Q, i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, such that

Giu⊥(x¯mi)=∑j∈Jj≠i∑v=0btv(xj)n−1,n−1q¯iujv(x¯mij),Hjv⊥(x¯mj)=−∑i∈Ii≠j∑u=0aq¯iujv(x¯mij)xi⌟tu (3.23)

for all x_m ∈ T_n(R)^m. Substituting (3.23) in (3.21), we get

∑i∈I∑u=0aGiu⊤(x¯mi)xi⌟tu+∑j∈J∑v=0b(tn−2′⌜)v(xj)n−2⌜Hjv⊤(x¯mj)−∑j∈J∑i∈Ii≠j∑v=0b∑u=0a(tn−2′⌜)v(xj)n−2,n−1⌝q¯iujv(x¯mij)xi⌟tu−∑j∈J∑i∈Ii≠j∑v=0b∑u=0atv′⌜¯(xj)n−1,n−1q¯iujv(x¯mij)xi⌟tu=0

for all x_m ∈ T_n(R)^m. That is,

∑i∈I∑u=0a(Giu⊤(x¯mi)−∑j∈Jj≠i∑v=0b(tn−2′⌜)v(xj)n−2,n−1⌝q¯iujv(x¯mij)−∑j∈Jj≠i∑v=0btv′⌜¯(xj)n−1,n−1q¯iujv(x¯mij))xi⌟tu+∑j∈J∑v=0b(tn−2′⌜)v(xj)n−2⌜Hjv⊤(x¯mj)=0

for all x_m ∈ T_n(R)^m. Using the same argument as (3.3) we can get from the induction hypothesis that there exist maps r_iujv : T_n(R)^m−2 → M_(n–2)×1(Q), i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, such that

Giu⊤(x¯mi)=∑j∈Jj≠i∑v=0b(tn−2′⌜)v(xj)n−2⌜r¯iujv(x¯mij)+∑j∈Jj≠i∑v=0b(tn−2′⌜)v(xj)n−2,n−1⌝q¯iujv(x¯mij)+∑j∈Jj≠i∑v=0btv′⌜¯(xj)n−1,n−1q¯iujv(x¯mij),Hjv⊤(x¯mj)=−∑i∈Ii≠j∑u=0ar¯iujv(x¯mij)xi⌟tu

for all x_m ∈ T_n(R)^m. Thus, we get

Giu⊤(x¯mi)Giu⊥(x¯mi)=∑j∈Jj≠i∑v=0btn−2′⌜tn−2,n−1′⌝0tv(xj)n−2⌜(xj)n−2,n−1⌝0(xj)n−1,n−1r¯iujv(x¯mij)q¯iujv(x¯mij)

and

Hjv⊤(x¯mj)Hjv⊥(x¯mj)=−∑i∈Ii≠j∑u=0ar¯iujv(x¯mij)q¯iujv(x¯mij)xi⌟tu

for all x_m ∈ T_n(R)^m. Define r_iujv : T_n(R)^m−2 → M_(n−1)×1(Q), i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, by

riujv(x¯mij)=r¯iujv(x¯mij)q¯iujv(x¯mij).

Hence, we have

Giu(x¯mi)=∑j∈Jj≠i∑v=0b(t′⌜)vxj⌜riujv(x¯mij),Hjv(x¯mj)=−∑i∈Ii≠j∑u=0ariujv(x¯mij)xi⌟tu

for all x_m ∈ T_n(R)^m.

According to the conclusions derived above, we get from (3.12) that λ_kuv = μ_kuv for all k ∈ 𝓘 ∩ 𝓙 and there exist maps r_iujv : T_n(R)^m−2 → M_(n−1)×1(Q), i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, such that

Eiu⌝(x¯mi)=∑j∈Jj≠i∑v=0b(t′⌜)vxj⌜riujv(x¯mij)+∑j∈Jj≠i∑v=0b(t′⌜)vxj⌝qiujv(x¯mij)+∑j∈Jj≠i∑v=0btv′¯xj⌟qiujv(x¯mij)+∑v=0bλiuv(x¯mi)tv′¯,Fjv⌝(x¯mj)=−∑i∈Ii≠j∑u=0ariujv(x¯mij)xi⌟(t′⌟)u−∑i∈Ii≠j∑u=0apiujv(x¯mij)xi⌜tu′¯−∑i∈Ii≠j∑u=0apiujv(x¯mij)xi⌝(t′⌟)u−∑u=0aλjuv(x¯mj)tu′¯λkuv=0ifk∉I∩J

for all x_m ∈ T_n(R)^m. Define P_iujv : T_n(R)^m−2 → T_n(Q), i ∈ 𝓘, j ∈ 𝓙, i ≠ j, 0 ⩽ u ⩽ a, 0 ⩽ v ⩽ b, by

Piujv(x¯mij)=piujv(x¯mij)riujv(x¯mij)0qiujv(x¯mij).

Now we can rewrite (3.1) as

Eiu(x¯mi)=∑j∈Jj≠i∑v=0bt′⌜t′⌝0t′⌟vxj⌜xj⌝0xj⌟piujv(x¯mij)riujv(x¯mij)0qiujv(x¯mij)+∑v=0bλiuv(x¯mi)00λiuv(x¯mi)t′⌜t′⌝0t′⌟v=∑j∈Jj≠i∑v=0b(t′)vxjPiujv(x¯mij)+∑v=0bλiuv(x¯mi)(t′)v,Fjv(x¯mj)=−∑i∈Ii≠j∑u=0apiujv(x¯mij)riujv(x¯mij)0qiujv(x¯mij)xi⌜xi⌝0xi⌟t′⌜t′⌝0t′⌟u−∑u=0aλjuv(x¯mj)00λjuv(x¯mj)t′⌜t′⌝0t′⌟u=−∑i∈Ii≠j∑u=0aPiujv(x¯mij)xi(t′)u−∑u=0aλjuv(x¯mj)(t′)u

for all x_m ∈ T_n(R)^m. Moreover, λ_kuv = 0 if k ∉ 𝓘 ∩ 𝓙. Thus, (3.2) has only a standard solution.
It remains to prove that the functional identity

∑i∈I∑u=0aEiu(x¯mi)xi(t′)u+∑j∈J∑v=0b(t′)vxjFjv(x¯mj)∈Z(Tn(Q))for allx¯m∈Tn(R)m

has only a standard solution, if max{∣𝓘∣ + a, ∣𝓙∣ + b} ⩽ d − 1.

The proof is similar to that of [14, Theorem 1.2].□

In particular, taking t = 1_Q and t′ = I_n in Theorem 1.2 we can obtain Theorem 1.1.

Acknowledgment

This research was supported by NNSF of China (No. 11771069).

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Received: 2019-09-16

Accepted: 2020-02-16

Published Online: 2020-03-20

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2020-0018

Keywords for this article

functional identity; (t; d)-free subset; upper triangular matrix ring

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