TEMA (São Carlos) vol.14 número2

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Half-Factoriality in Subrings of Trigonometric Polynomial Rings

EHSAN ULLAH* and TARIQ SHAH

Received on October 26, 2012 / Accepted on April 20, 2013

ABSTRACT.Trigonometric polynomials are widely used in different fields of engineering and science. Inspired by their applications, we investigate half-factorial domains in trigonometric polynomial rings. We construct the half-factorial domainsT₂′,T₃′andT₄′which are the subrings of the ring of complex trigono-metric polynomialsT′, such thatT₂′ _⊆ T₃′ _⊆ T₄′ _⊆ T′.We also discuss among these three subrings the

Condition: LetA_⊆ Bbe a unitary (commutative) ring extension. For eachx _∈ Bthere existx′ _∈U(B) andx′′_∈Asuch thatx₌x′x′′, whereU(B)denote the group of units ofB.

Keywords:trigonometric polynomial, HFD, condition 1, condition 2, irreducible.

Mathematical subject classification:13A05, 13B30, 12D05, 42A05.

1 INTRODUCTION

Often it becomes interesting and fruitful to bridge different fields of study. Some time resolving one problem becomes much easier if we translate it into some equivalent problem. As we know from computer science, all NP-hard problems are equivalent. Resolution of one NP-hard problem can benefit the other, if we are able to translate the problem correctly. For example, logical solvers like SAT Solvers are used for cryptanalysis and solving algebraic system of equations over GF(2) [3, 4]. The more recent fields of study in this direction are, approximate commutative algebra [18] and numerical algebraic geometry [22]. We are hopeful that our study in this paper could be one of such contributions.

Trigonometric polynomials are widely used in different fields of engineering and science, like trigonometric interpolation applied to the interpolation of periodic functions, approximation the-ory, discrete Fourier transform, and real and complex analysis, etc. We are developing this study by keeping in mind, the possibility that studding factorization properties of these polynomials could help studying the above fields and especially Fourier series, that is, the study of big waves (a trigonometric polynomial) in terms of small wavelets (irreducibles). The study of Fourier se-ries is a vast field of study by itself and this study will help to understand a big Fourier sese-ries in terms of smaller Fourier series.

*Corresponding author

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We refer to [9, 10] and reference therein, for a short review of some of the recent interesting results on nonnegative trigonometric polynomials and their applications in Fourier series, sig-nal processing, approximation theory, function theory and number theory. Many applications, especially in mechanical engineering and in numerical analysis lead to quantifier elimination problems with trigonometric functions involved (see [14]). Decompositions of trigonometric polynomials with applications to multivariate subdivision schemes is studied in [11], random almost periodic trigonometric polynomials and applications to ergodic theory can be found in [6], a detailed treatment of trigonometric series can be found in [26], and a new proof of a theo-rem of Littlewood concerning flatness of unimodular trigonometric polynomials is given in [5], this proof is shorter and simpler than Littlewood’s. Inspired by the above stated applications and lot more, we investigate trigonometric polynomials using an algebraic approach. Throughout this article we follow the notation and definitions introduced in [15, 25] unless mentioned otherwise.

In polynomial rings, factorization properties of integral domains have been a frequent topic of recent mathematical literature. Recall that a Unique Factorization Domain (UFD) is an integral domain in which every non-unit element can be uniquely expressed, up to isomorphism, as a product of irreducible elements. A Principal Ideal Domain (PID) is an integral domain in which every ideal is principal, i.e., can be generated by a single element. An integral domainDis atomic if each nonzero nonunit ofDis a product of irreducible elements(atoms)ofD, [7] and it is well known that UFDs, PIDs and Noetherian domains are atomic domains. In an integral domainD, if there does not exist any infinite strictly ascending chain of principal integral ideals, it is said to satisfy theascending chain condition on principal ideals (ACCP).Noetherian domains, PIDs and UFDs satisfy ACCP and domains satisfying ACCP are atomic. For examples of atomic domains which do not satisfy ACCP we refer to Grams [12] and Zaks [24].

Another interesting type of domains in commutative algebra is of half-factorial domains. An integral domain D is said to be a half-factorial domain (HFD) if D is atomicand whenever

x1. . .xm = y1. . .yn, wherex1,x2. . .xm, y1,y2. . .ynare irreducibles in D, thenm=n [23].

Since we obtain HFDs by dropping uniqueness condition on UFDs, a UFD is obviously an HFD. But the converse is not true, since any Krull domain DwithC I(D) ∼₌Z2is an HFD [23], but

not a UFD.Moreover, a polynomial extension of an HFD is not an HFD anymore, for instance

Z_[√₋3][X_]is not an HFD, asZ_[√₋3]is an HFD but not integrally closed [8]. In general we have,

UFD_=⇒HFD_=⇒ACCP_=⇒Atomic

Note that none of the above implications is reversible. ByU(D)we denote the group of units ofD.

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For the factorization of exponential polynomials, J.F. Ritt developed: “If

1+a1eα1x+ · · · +aneαnx is divisible by 1+b1eβ1x+ · · · +breβrx

with nob ₌0, then everyβ is a linear combination ofα1,..., αnwith rational coefficients” [17,

Theorem]. Latter on getting inspired by this, G. Picavet and M. Picavet [15] investigated some factorization properties in trigonometric polynomial rings. Following [15], when we replace all

αkabove byim, withm∈Z, we obtain trigonometric polynomials. Whereas

T′ ₌

n

k=0

(akCoskx ₊bkSinkx) _: n_∈N, ak,bk _∈C

and

T ₌

n

k=0

(akCoskx +bkSinkx) : n∈N, ak,bk ∈R

are the trigonometric polynomial rings.

Again following [15],Sin2x₌(1−Cosx)(1+Cosx)shows that two different non-associated irreducible factorizations of the same element may appear. Throughout we denote byCoskx

andSinkx the two functionsx _→Coskx andx _→ Sinkx (defined overR). Also from basic trigonometric identities, it is obvious that for eachn_∈N_\{1},Cosnx represents a polynomial in

Cosx with degreenandSinnx represents the product ofSinxand a polynomial inCosx with degreen₋1.Conversely by linearization formulas, it follows that any productCosnx Sinpxcan be written as:

q

k=0

(akCoskx ₊bkSinkx), where q _∈Nandak,bk_∈Q.

HenceT ₌R_[Cosx,Sinx_{] ⊆}C_[Cosx,Sinx_{] =}T′.

As proved in [15, Theorem 2.1 & Theorem 3.1],T′is a Euclidean domain andT is a Dedekind half factorial domain. We continue the investigations to find the factorization properties in trigonometric polynomial rings, begun in [15] and extended in [20] and [21], by extending this study towards finding half-factorial subrings of the complex trigonometric polynomial ringT′. In section 2, we define three subringsT₂′,T₃′andT₄′ofT′. We exploreT₂′,T₃′andT₄′, and prove that each of the three new subrings,T₂′,T₃′, andT₄′ is an HFD isomorphic to (Q₊XR_[X_])X,

(Q₊XC_[X_])X and(Q(i)+XC[X])X respectively. In section 3, we discussed Condition 1 (see

[16, page 661]) among the three half-factorial domains. We conclude this paper by summarizing the results and discussing further research and applications.

2 THE SUBRINGS OFC_[Cosx,Sinx_]

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2.1 The Construction ofT₂′

Consider the following set

T₂′₌

n

k=0

(akCoskx ₊ibkSinkx), n_∈N, ak,bk_∈R, an₌α₊β,bn₌β₋α

,

whereα_∈Q,β_∈Randα, βare not simultaneously zero. Letz_∈ T₂′.We may write

z₌a0+

n−1

k=1

(akCoskx ₊ibkSinkx)_{+ {}(α₊β)Cosnx ₊i(β₋α)Sinnx_}.

AsCosx ₌eix+₂e−ix andSinx₌eix−₂_ie−ix, we have

z₌e−inx

a0+

n−1

k=1

a_k+b_k

2

ei(n+k)x₊ak−bk

2

ei(n−k)x ₊βei2nx ₊α

,

whereak+bk

2 ,

ak−bk

2 , α0, β∈R,α∈Q. Sincezis an arbitrary, every element ofT2′is of the form

e−inxP(ei x), n_∈N, whereP(X)_∈Q₊XR_[X_].

Conversely, forα0∈Qandαk ∈R, 1≤k≤2n, we have

e−inxP(ei x)₌α0e−inx +α2neinx +

n−1

k=1

αke−i(n−k)x ₊α2n−kei(n−k)x+αn.

Asei x ₌Cosx₊i Sinx, we have

e−inxP(ei x) ₌ (α0+α2n)Cosnx+i(α2n−α0))Sinnx

+

n−1

k=1

{(αk+α2n−k)Cos(n −k)x

+i(α2n−k −αk))Sin(n −k)x} +αn,

whereαn, α0+α2n, α2n−α0, αk+α2n−k, α2n−k−αk ∈R. Therefore every element which is

of the forme−inxP(ei x),n_∈N, whereP(X)_∈Q₊XR_[X_], is inT₂′.

Conclusion 1. The consequence of above construction is: T₂′ _{= {}e−inxP(ei x),n _∈ N, where

P(X) ∈ Q₊XR_[X_]. So we have an isomorphism f _: (Q₊XR_[X_])X −→ T₂′ through the

substitution morphismX _−→ei x. ThereforeT₂′_⋍(Q₊XR_[X_])X.

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Proof. SinceX is a prime inQ₊XR_[X_][1, Example 1.8 (b)], so(Q₊XR_[X_])_X is a local-ization ofQ₊XR_[X_]by a multiplicative system generated by a prime. AlsoQ₊XR_[X_]is an HFD [2, Proposition 3.1]. Therefore(Q₊XR_[X_])_X is an HFD [1, Corollary 2.5]. Hence the isomorphismT₀′_⋍(Q₊XR_[X_])Xin Conclusion 1 gives the result.

2.2 The Construction ofT₃′

We define the setT₀′of all polynomials of the form

n

k=0

(akCoskx +bkSinkx), n∈N, ak,bk ∈C and

an=α+γ+iβ,bn= −β+i(α−γ ),

such thatα, β∈R,γ ∈Q, whereα, βandγ are not simultaneously zero.

Letz_∈T₃′be an arbitrary element, so we may write

z ₌ a0+

n−1

k=1

(akCoskx ₊bkSinkx)₊

(α₊γ₊iβ)Cosnx₊(₋β₊i(α₋γ ))Sinnx

.

AsCosx ₌ eix+₂e−ix andSinx ₌eix−₂e_i−ix, so

z ₌ a0+

n−1

k=1

a′

k+b′′k+i(a′′k −bk′)

2

eikx ₊a

′

k−b′′k+i(ak′′+bk′)

2

e−ikx

+(α₊iβ)einx ₊γe−inx,

whereak=a_k′ +ia_k′′,bk=b′_k+ib_k′′ and a′_k,a_k′′,b′_k,b_k′′∈R,a0∈C.

Setting

α_k′ ₌a ′

2 and β

′

k =

a_k′ ₋b′′_k₊i(a′′_k ₊b′_k)

2 ,

we have

z₌e−inx

a0einx +

n−1

k=1

α_k′ei(n+k)x₊β_k′ei(n−k)x

+(α₊iβ)ei2nx ₊γ

,

whereα′_k,β_k′,a0∈C, andα, β∈R,γ ∈Q.

Sincezis an arbitrary, every element ofT₃′is of the form

e−inxP(ei x), n _∈N, where P(X)_∈Q₊XC_[X_].

Conversely, forα0∈Qandαk∈C, 1≤k≤2n, we have

e−inxP(ei x)₌α0e−inx +α2neinx+ n−1

k=1

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Asei x ₌Cosx₊i Sinx, so by setting

αk =αk′ +iα′′k, α2n−k =α′2n−k+iα2′′n−k and α2n=α′2n+iα2′′n.

We have

e−inxP(ei x) ₌ (α0+α′₂_n+iα₂′′_n)Cosnx+(−α′′₂_n+i(α₂′_n−α0))Sinnx

+

n−1

k=1

{(α_k′ ₊α₂′_n₋_k₊i(α_k′′₊α′′₂_n₋_k))Cos(n ₋k)x

+(α_k′′₋α′′₂_n₋_k₊i(α′₂_n₋_k₋α_k′))Sin(n₋k)x_{} +}αn.

= anCosnx +bnSinnx+ n−1

k=1

akCos(n−k)x+bkSin(n−k)x+αn,

where

an ₌ α0+α2′n+iα2′′n, bn = −α2′′n+i(α′2n−α0),

ak ₌ α_k′ ₊α₂′_n₋_k₊i(α′′_k ₊α₂′′_n₋_k) and bk ₌ α_k′′₋α′′₂_n₋_k₊i(α₂′_n₋_k₋α_k′).

Therefore every element which is of the forme−inxP(ei x),n _∈N, whereP(X)_∈Q₊XC_[X_], is inT₃′.

Conclusion 2. The consequence of above construction is: T₃′ ₌ e−inxP(ei x),n _∈ N, where

P(X) ∈ Q₊XC_[X_]

. So we have an isomorphism f _: (Q₊XC_[X_])X −→ T₃′through the

substitution morphismX _−→ei x. ThereforeT₃′_⋍(Q₊XC_[X_])X.

Theorem 2.The integral domain T₃′is an HFD having irreducible elements, up to units, trigono-metric polynomials of the form Cosx ₊i Sinx₋a, where a_∈C_\{0}.

Proof. Since X is a prime inQ₊XC_[X_][1, Example 1.8 (b)], so(Q₊XC_[X_])X is a

local-ization ofQ₊XC_[X_]by a multiplicative system generated by a prime. AlsoQ₊XC_[X_]is an HFD [2, Proposition 3.1]. Therefore(Q₊XC_[X_])_X is an HFD [1, Corollary 2.5]. Now use the isomorphismT′

3⋍(Q+XC[X])X in Conclusion 2.

2.3 The Construction ofT₄′

Consider the setT₄′of polynomials of the form

n

k=0

(akCoskx ₊bkSinkx), n_∈N, ak,bk_∈C and

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such thatα, β_∈R,γ , δ_∈Q, whereα, β,γ andδare not simultaneously zero. Letz_∈T₄′be an arbitrary element, so we may write it as

z ₌ a0+

n−1

k=1

(akCoskx +bkSinkx)

+(α₊γ )₊i(β₊δ)Cosnx ₊(δ₋β)₊i(α₋γ )Sinnx.

AsCosx ₌ eix+₂e−ix andSinx ₌eix−₂e_i−ix, so

z ₌ a0+

n−1

k₌1

a′

2

eikx ₊a

′

k−b′′k+i(ak′′+bk′)

2

e−ikx

+(α₊iβ)einx ₊(γ₊iδ)e−inx,

whereak=a_k′ +ia_k′′,bk=b′_k+ib_k′′ and a′_k,a_k′′,b′_k,b_k′′∈R,a0∈C.

Setting

α_k′ ₌a ′

2 and β

′

k =

a_k′ ₋b′′_k₊i(a′′_k ₊b′_k)

2 ,

we have

z₌e−inx

a0einx +

n−1

k=1

{α_k′ei(n+k)x ₊β_k′ei(n−k)x_{} +}(α₊iβ)ei2nx ₊(γ ₊iδ)

,

whereα′_k,β_k′,a0∈C, andα, β∈R,γ , δ∈Q.

Sincezis an arbitrary, every element ofT₄′is of the form

e−inxP(ei x), n_∈N, whereP(X)_∈Q(i)₊XC_[X_].

Conversely, forα0∈Q(i)andαk ∈C,1≤k≤2n, consider

e−inxP(ei x) ₌ α0e−inx +α2neinx + n−1

k=1

αke−i(n−k)x +α2n−kei(n−k)x+αn.

Asei x ₌Cosx ₊i Sinx, so for

α0=α₀′ +iα₀′′, αk =α_k′ +iα′′_k, α2n−k =α₂′_n₋_k+iα₂′′_n₋_k and α2n=α₂′_n+iα₂′′_n.

We have

e−inxP(ei x) ₌ (α0+α2′n+i(α0′′+α2′′n))Cosnx +(α0′′−α2′′n+i(α2′n−α0))Sinnx

+

n−1

k=1

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+(α_k′′₋α′′₂_n₋_k₊i(α₂′_n₋_k₋α′_k))Sin(n₋k)x_{} +}αn.

= anCosnx +bnSinnx+ n−1

k=1

{akCos(n −k)x+bkSin(n−k)x} +αn,

where

an ₌ α0+α2′n+i(α0′′+α2′′n), bn = α0′′−α′′2n+i(α′2n−α0),

ak ₌ α_k′ ₊α₂′_n₋_k₊i(α′′_k ₊α₂′′_n₋_k) and bk ₌ α_k′′₋α′′₂_n₋_k₊i(α₂′_n₋_k₋α_k′).

Therefore every element which is of the forme−inxP(ei x),n_∈N, whereP(X)_∈Q(i)₊XC_[X_], is inT₄′.

Conclusion 3. The consequence of above construction is: T₄′ _{= {}e−inxP(ei x),n _∈ N, where

P(X)_∈Q(i)₊XC_[X_]}. So once again we have an isomorphism f _:(Q(i)₊XC_[X_])X −→T₄′

through the substitution morphismX_−→ei x. ThereforeT₄′_⋍(Q(i)₊XC_[X_])X.

Theorem 3.The integral domain T₄′is an HFD having irreducible elements, up to units, trigono-metric polynomials of the form Cosx ₊i Sinx₋a, where a_∈C_\{0}.

Proof. The proof follows analogous to Theorem 1 and 2.

The following is the analogue of [15, Corollary 2.2] and gives the factorization in T₄′ instead ofT′.

Corollary 1.Let z₌ n

k=0

(akCoskx +bkSinkx), n∈N\{1}, ak,bk ∈Cwith(an,bn)=(0,0),

such that an =(α+γ )+i(β+δ)and bn =(δ−β)+i(α−γ ), whereα, β ∈R,γ , δ ∈Q. Let d be a common divisor of the integers k such that(ak,bk) = (0,0). Then z has a unique factorization

λ(Cosnx₋i Sinnx)

2n d

j=1

(Cosd x ₊i Sind x₋αj), where λ, αj ∈C\{0}.

Proof. SinceT₄′_⊆T′, therefore proof follows by [15, Corollary 2.2].

Remark 1.The factorization inT₂′andT₃′is an analogue of Corollary 1.

Now onwards the symbol∩in all diagrams will represent the inclusion⊆.

Remark 2.Consider the following ascending chain of substructures ofC_[X_]

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whereQ₊XR_[X_],Q₊XC_[X_]andQ(i)₊XC_[X_]are HFDs, whereasC_[X_]is a Euclidean do-main. The localization of all these four by a multiplicative system generated byXpreserves their factorization properties.

Q₊XR_[X_] _⊆ Q₊XC_[X_] _⊆ Q(i)₊XC_[X_] _⊆ C_[X_]

∩ ∩ ∩ ∩

(Q₊XR_[X_])X ⊆ (Q+XC[X])X ⊆ (Q(i)+XC[X])X ⊆ C[X]X.

Using Conclusion 1, Conclusion 2, Conclusion 3 and [15, Theorem 2.1], we have

Q₊XR_[X_{] ⊆} Q₊XC_[X_{] ⊆} Q(i)₊XC_[X_{] ⊆} C_[X_]

∩ ∩ ∩ ∩

T₂′ _⊆ T₃′ _⊆ T₄′ _⊆ T′,

whereT₂′,T₃′,T₄′are HFDs contained in the Euclidean domainT′. So we can see that there is a ascend of factorization properties for each of these four structures.

Remark 3.Due to the domain extensions

Q₊XR_[X_] _⊆ (Q₊XR_[X_])X, Q+XC[X] ⊆ (Q+XC[X])X

Q(i)₊XC_[X_{] ⊆} (Q(i)₊XC_[X_])X and C[X] ⊆ (C[X])X,

we obtain the following four extended ideals.

1. Consider the domain extensionC_[X_{] ⊆} (C_[X_])X. Since XC[X]is a maximal ideal of

C_[X_]andXC_[X_{] ∩}(X)₌φ, the extended ideal(XC_[X_])e₌(C_[X_])X [25, Corollary 2].

Hence(XC_[X_])e⋍T′_{by [15, Theorem 2.1].}

2. If we consider the domain extension Q₊XR_[X_{] ⊆} (Q₊XR_[X_])X, we observe that XR_[X_]is a maximal ideal ofR_[X_]andXR_[X_{] ∩}(X)₌φ. Therefore the extended ideal

(XR_[X_])e₌(Q₊XR_[X_])X[25, Corollary 2]. Hence(XR[X])e ⋍T₂′by Conclusion 1.

3. Analogous to (2) we have(XC_[X_])e ⋍T₃′using Conclusion 2, sinceXC_[X_]is a maximal ideal ofC_[X_]for the domain extensionQ₊XC_[X_{] ⊆}(Q₊XC_[X_])X.

4. Once again analogous to (2) and (3) we have(XC_[X_])e _⋍T₄′using Conclusion 3, since

XC_[X_]is a maximal ideal ofC_[X_]for the domain extensionQ(i)₊XC_[X_{] ⊆}(Q(i)₊

XC_[X_])X.

Definition 1.We denote the set T₂′forα₌0by I2and is defined as

I2=

n

k=0

(akCoskx +ibkSinkx) : n∈N, ak,bk∈R and an=bn=β

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Definition 2.We denote the set T₃′forγ ₌0by I3and is defined as

I3=

n

k=0

(akCoskx +bkSinkx) : n ∈N, ak,bk ∈C and an=α+iβ,bn= −β+iα

.

Definition 3.We denote the set T₄′forγ ₌δ₌0by I4and is defined as

I4=

n

k₌0

(akCoskx ₊bkSinkx) _: n _∈N, ak,bk _∈C and an₌α₊iβ,bn_{= −}β₊iα

.

Remark 4. By Definition 2 and Definition 3, we have I3 = I4, so onwards we useI to denote

I3andI4.

Lemma 1. For the maximal ideal XR_[X_](respectively XC_[X_]) ofQ₊XR_[X_](respectively of

Q₊XC_[X_]andQ(i)₊XC_[X_]), we have(XR_[X_])_X _⋍I2(respectively(XC[X])X ⋍I ).

Proof. Follows by Conclusion 1 (respectively Conclusion 2 and Conclusion 3).

3 CONDITIONS SATISFIED BY TRIGONOMETRIC POLYNOMIAL RING EXTENSIONS

In this section we study two special conditions among trigonometric polynomial ring extensions. First one known as Condition 1, is borrowed from [16, page 661] and the second one is borrowed from [20]. We denote the group of units of a ringBbyU(B).

Condition 1.Let A_⊆B be a unitary (commutative) ring extension. For each x _∈ B there exist x′_∈U(B)and x′′_∈ A such that x ₌x′x′′.

Example 1.Following [16, Example 1.1].

a) If the ring extensionA_⊆Bsatisfies Condition 1, the ring extensionA₊X B_[X_{] ⊆}B_[X_]

(orA₊X B_[[X_{]] ⊆}B_[[X_]]) also satisfies Condition 1.

b) If the ring extensions A_⊆ B andB _⊆C satisfy Condition 1, so does the ring extension

A_⊆C.

c) IfBis a fraction ring ofA, the ring extensionA_⊆Bsatisfies Condition 1. Hence the ring extensionA_⊆Bsatisfying Condition 1 is a generalization of localization.

d) IfBis a field, the ring extension A_⊆Bsatisfies Condition 1.

Condition 2.Let A_⊆B be a unitary (commutative) ring extension and let A1⊆B1be a unitary

(commutative) ring extension where A _⊆ A1 and B ⊆ B1. Then for each x ∈ B1 there exist

x′_∈U(B)and x′′_∈ A1such that x=x′x′′.

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Proof. Since the ring extension A _⊆ B satisfies Condition 1, for each a _∈ B there exist

b_∈U(B)andc_∈ Asuch thata ₌bc. ObviouslyN−1A_⊆N−1B. Letx₌ a_s _∈ N−1B. Then

x₌ a_s,a _∈B,s_∈ N.This impliesx₌ bc_s ₌bc_s, whereb_∈U(B)andc_s ∈N−1A.

The converse of Lemma 2 does not hold in general. The following proposition addresses this fact.

Proposition 1.Let A_⊆B be a unitary (commutative) ring extension. Let the domain extension A1 ⊆ B1satisfies Condition2and the domain extension A ⊆ A1 satisfies Condition1, where

A_⊆A1and B⊆B1. If U(B1)=U(B)and U(B1)∩A1=U(A1). Then the domain extension

A_⊆B satisfies Condition1.

Proof. Since the domain extensionA1 ⊆B1satisfies Condition 2, for eachx ∈ B⊆ B1there

existx₁′ _∈U(B)andx₁′′ _∈ A1such thatx =x′₁x′′₁. Now ifx₁′′ ∈ A, there is nothing to prove. If

x₁′′ _∈ A1\A, we proceed as follows. Since the domain extensionA ⊆ A1satisfies Condition 1,

forx₁′′_∈ A1there existx₂′ ∈U(A1)andx′′₂∈ Asuch thatx′′₁=x₂′x₂′′. It follows thatx =x₁′x′₂x′′₂.

AsU(B1)=U(B)andU(B1)∩A1 =U(A1), sox₁′x′₂∈U(B)andx₂′′∈ A. Therefore A⊆B

satisfies Condition 1.

Proposition 2. Let A _⊆ B _⊆ B1 be a unitary (commutative) ring extension. Assume that the

domain extension A _⊆ B1satisfies Condition2. Then the domain extension A ⊆ B satisfies

Condition1.

Proof. The domain extensionA_⊆B1satisfies Condition 2, so for eachx∈ B⊆B1there exist

x′_∈U(B)andx′′_∈ Asuch thatx₌x′x′′. Therefore A_⊆Bsatisfies Condition 1.

Example 2. Let A_⊆ B be a unitary (commutative) ring extension. Let N ₌U(A)is a multi-plicative system inA. Assume that the domain extensionN−1A_⊆N−1Bsatisfies Condition 2. Then the domain extensionA_⊆Bsatisfies Condition 1.

Example 3.Some more interesting examples satisfying Condition 1 are as follows.

(a) If the ring extensionsA _⊆ B andB _⊆Csatisfy Condition 2, so does the ring extension

A_⊆C.

(b) For A₌A1andB=B1the Condition 1 and Condition 2 coincides.

(c) If the ring extensionA1⊆B1satisfies Condition 2, it does satisfies Condition 1.

(d) By Lemma 2, the ring extensionsT₂′ _⊆T₃′,T₃′_⊆T₄′, andT₄′ _⊆T′satisfy Condition 2 so do the ring extensionsT′

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3.1 Conclusion

Consider the commutative inclusion diagram due to Remark 2.

Q₊XR_[X_{] ⊆} Q₊XC_[X_{] ⊆} Q(i)+XC_[X_{] ⊆} C_[X_]

∩ ∩ ∩ ∩

T₂′ _⊆ T₃′ _⊆ T₄′ _⊆ T′

The following table concludes our discussion on Condition 1 and Condition 2 among trigono-metric polynomial ring extensions.

Ring Extension Condition 1 Condition 2

Q₊XC_[X_{] ⊆}C_[X_] Yes Yes

Q(i)₊XC_[X_{] ⊆}C_[X_] Yes Yes

Q₊XR_[X_{] ⊆}T₂′ Yes Yes

Q₊XR_[X_{] ⊆}T₃′ Yes Yes

Q₊XR_[X_{] ⊆}T₄′ Yes Yes

C_[X_{] ⊆}T′ Yes Yes

T₂′_⊆T₃′ No Yes

T₃′_⊆T₄′ No Yes

T₄′_⊆T′ No Yes

Q₊XR_[X_{] ⊆} T₃′ No Yes

Q₊XC_[X_{] ⊆} T₄′ No Yes

Q(i)₊XC_[X_{] ⊆} T′ No Yes

T₂′_⊆T₄′ No Yes

T₂′_⊆T′ No Yes

T₃′_⊆T′ No Yes

Q₊XR_[X_{] ⊆}Q₊XC_[X_] No No

Q₊XC_[X_{] ⊆}Q(i)₊XR_[X_] No No

Note that the extensionsQ₊XR_[X_{] ⊆}T₃′,Q₊XC_[X_{] ⊆}T₄′,Q(i)₊XC_[X_{] ⊆}T′,T₂′ _⊆T₄′,

T₂′_⊆T′andT₃′_⊆T′satisfy Condition 2 due to transitivity.

3.2 Future Work and Applications

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factor-ization properties of trigonometric polynomial rings and this can open a new challenge for the researchers.

In addition to the applications mentioned in the introduction, we would like to highlight an ap-plication of trigonometric polynomials in symbolic computation. In [13], J. Mulholland and M. Monagan presented algorithms for simplifying ratios of trigonometric polynomials and algo-rithms for dividing, factoring and computing greatest common divisors of trigonometric polyno-mials. The provided algorithms do not always lead to the simplest form. A possible direction of study could be to provide enough general algorithms for finding a simplest form.

ACKNOWLEDGMENTS

The authors would like to thank referees for deep and valuable discussions on the subjects of this paper.

RESUMO. Polinômios trigonométricos são amplamente utilizados em diferentes áreas de engenharia e ciências. Inspirado por suas aplicações, investigamos dom´ınios semi-fatorial em

anéis de polinômios trigonométricos. Nós constru´ımos o dom´ınios semi-fatoriais T₂′, T₃′ e

T₄′que são os subanéis do anel de complexos, polinômios trigonométricasT′, de tal forma queT₂′ _⊆T₃′ _⊆ T₄′ _⊆ T′. Discutimos, também, entre estes três subanéis aCondição: Seja

A_⊂ Bser uma extens˜ao unit´aria (comutativa) do anel. Para cadax _∈Bexistemx′_∈U(B)

ex′′_∈ Atal quex₌x′x′′, ondeU(B)´e o grupo de unidades deB.

Palavras-chave:polinômios trigonométricos, HFD, condição 1, condição 2, irredut´ıvel.

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