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J. Math. Comput. Sci. 7 (2017), No. 1, 30-38 ISSN: 1927-5307

THE ORDER OF THE SECOND GROUP OF UNITS OF THE RING Z[i]/ <β >

WIAM M. ZEID

Department of Mathematics, Lebanese International University, Saida, Lebanon

Copyright c2017 Wiam M. Zeid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. In this article we introduce a newely defined functionφ2G(β)that represents the order of the second

group of units of the ringR=Z[i]/ <β >. We examine some of the properties of this function that are similar to that of the Euler Phi functionφ(n).

Keywords: commutative ring; Euler Phi function; Gaussian primes; group of units; generalized group of units;

ring of Gaussian integers.

2010 AMS Subject Classification:11R04, 13F15, 16U60.

1.

Introduction

The Euler Phi function,φ(n),has been generalized through different approaches and was

stud-ied in many domains. In 1983, J.T Cross [1], extended the defintion of the Euler Phi function to the domain of Gaussian integersZ[i], whereφG(β)represents the order of the group of units

of the ring Z[i]/ <β > where β is a non zero Gaussian integer. That is φG(β) =|U(β)|.

Cross gave a complete charecterization for the group of unitsΦG(β)as shown in the following

theorem.

Corresponding author

Received August 23, 2016

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Theorem 1.1.

(1) U(πn)∼=Zqnqn−1.

(2) U(pn)∼=Zpn−1×Zpn−1×Zp21.

(3) U(1+i)∼=Z1.

(4) U((1+i)2)∼=Z2.

(5) U((1+i)n)∼=Z2m−1×Z2m−2×Z4ifn=2m.

(6) U((1+i)n)∼=Z2m−1×Z2m−1×Z4ifn=2m+1.

In 2006, a generalization for the group of units of any finite commutative ring with identity was introduced by El -Kassar and Chehade [2]. They proved that the group of units of a commu-tative ringRwith identity,U(R),supports a ring structure and this made it possible to define the second group of units ofRasU2(R) =U(U(R)).Extending this definition to thekthlevel, the kthgroup of units is defined asUk(R) =U(Uk−1(R)).This generalization allowed to formulate a new generalization to the Euler Phi function that represents the order of the generalized group of units of the ringRand is denoted byφk(R).In [4], Assaf gave an explicit formula,φ2(R), for

the order of the second group of units in the domain of integers.

In this paper, we introduce a newly defined function, φ2G(β), the order of the second group

of unitsU2(β) =Φ2G(β)of the ringZ[i]/ <β > .

Below we state two theorems that were discussed in [2].

Theorem 1.2. If a group(G,∗)is isomorphic to the additive group of a ring(R,+, .), then there is an operation⊗on G such that(G,∗,⊗)is a ring isomorphic to(R,+, .).

Theorem 1.3. If R∼=R1⊕R2⊕...⊕Ri, then the group of units U(R)and U(R1)×U(R2)×...×

U(Ri)are isomorphic.

In [4], Assaf gave the definition of the second group of units in the domain of integers and listed some related properties.

Throughout this paper,

• nis a positive integer.

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• β andγ always denote any non- zero, non unit Gaussian integers.

• pand pjalways denote prime integers that are congruent to 3 modulo 4.

• π andπj always denote Gaussian prime integers of the form a+bi whereaandbare

non zero integers.

• q=π π =a2+b2.

• qandqjalways denote prime integers that are congruent to 1 modulo 4.

2.

Definition of

φ

2G

(

β

)

In this section, we define the new functionφ2G(β)and prove its multilplivcate property.

Definition 2.1. Defineφ2G(β)to be the order of the second group of units U2(Z[i]/ <β >) = Φ2G(β).Thus we write,φ2G(β) =|Φ2G(β)|.

Lemma 2.1. The functionφ2Gis multiplicative.

Proof. Let gcd(β,γ)1,thenZ[i]/ <β γ >∼=Z[i]/ <β >⊕Z[i]/ <γ > . Applying theorem

1.3 fork=2 andR=Z[i]/ <β γ >we get,

U2(Z[i]/ <β γ >)∼=U2(Z[i]/ <β >)×U2(Z[i]/ <γ >).Then

|U2(Z[i]/ <β γ >)|=|U2(Z[i]/ <β >)|.|U2(Z[i]/ <γ >)|.Consequently,

φ2G(β γ) =φ2G(β)φ2G(γ).

The fact theφ2Gis a multiplicative function gives a very important step in finding a general

formula of the order of the second group of units.

Lemma 2.2. Ifβ =α orβ =α2, thenφ2G(β) =1, elseφ2G(β)is even.

Proof. Ifβ =α orβ =α2, thenU2(β)is trivial, see [3]. Hence|U2(β)|=φ2G(β) =1.Now,

suppose thatβ 6=α andβ 6=α2, then applying the fundemental theorem of abelian groups we

get,U(β)∼=Zn1×Zn2×...×Znk and in ring structureU(β)∼=Zn1⊕Zn2⊕...⊕Znk. Moreover,

U2(β)∼=U(Zn1)×U(Zn2)×...×U(Znk).Thenφ

2

G(β) =φ(n1)φ(n2)...φ(nk)a product of even

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3.

The Explicit formula for

φ

2G

(

β

)

In order to determine an explicit formula forφ2G, we have to examine the cases whenβ =αn, πn,and pn.

Lemma 3.1. Let n be a positive integer, then

φ2G(αn) = 

   

   

1 if n≤2 2 if 3≤ n≤4 2n−4 if n≥5.

Proof. Letn≥5, then using theorem 1.1 we have

ΦG(αn)∼= 

Z2m−1×Z2m−2×Z4i f n=2m.

Z2m−1×Z2m−1×Z4i f n=2m+1.

Ifn=2m,Φ2G(αn)∼=U(Z2m−1)×U(Z2m−2)×U(Z4).

Hence,

|Φ2G(αn)|=|U(Z2m−1)|.|U(Z2m−2)|.|U(Z4)|=φ(2m−1).φ(2m−2).φ(4).

Consequently,φ2G(αn) =2n−4.

Ifn=2m+1, thenΦG2(αn)∼=U(Z2m−1)×U(Z2m−1)×U(Z4). So,

|Φ2G(αn)|=|U(Z2m−1)|.|U(Z2m−1)|.|U(Z4)|=φ(2m−1).φ(2m−1).φ(4).

Hence,φ2G(αn) =2n−4.

Ifn=1 or 2, thenΦ2G(αn)is trivial. Therefore,φ2G(αn) =1.

Ifn=3,ΦG(αn)∼=Z4.Thenφ2G(αn) =|Φ2G(αn)|=|U(Z4)|=2.

Ifn=4,ΦG(αn)∼=Z2×Z4.This givesφ2G(αn) =|Φ2G(αn)|=|U(Z2)|.|U(Z4)|=2.

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Proof. Using theoerm 1.1, we haveΦG(πn)=∼Zqnqn−1. Then,Φ2G(πn)∼=U(Zqnqn−1).

There-fore,|U(Zqnqn−1)|=φ(qn−qn−1) =φ(qn−1(q−1)).Consequently,φ2G(πn) =φ(φG(πn)).

Lemma 3.3. φ2G(pn) = 

φ(φG(p)), if n=1

(p−1)2p2n−4φ(φG(pn)), if n≥2.

Proof. Theorem 1.1 givesΦG(p)∼=Zp21whenn=1. Thus,ΦG2(p)∼=U(Zp21)and|Φ2G(p)|=

|U(Zp21)|.Consequently,φG2(p) =φ(p2−1) =φ(φG(p)).

Forn≥2,we haveΦG(pn)∼=Zpn−1×Zpn−1×Zp21.Thus,

Φ2G(p)∼=U(Zpn−1)×U(Zpn−1)×U(Zp21)

and|Φ2G(p)|=|U(Zpn−1)|.|U(Zpn−1)|.|U(Zp21)|.Consequently,

φ2G(pn) =φ(pn−1)φ(pn−1)φ(p2−1) = (p−1)2p2n−4φ(φG(p)).

Notation 3.1. For simplicity, we denote by fp(n)to be a function defined as

fp(n) =

1 if n=1

(p−1)2p2n−4if n≥2.

Using this notation, we can writeφ2G(pn) = fp(n)φ(φG(pn)).

Following lemmas 3.1,3.2 and 3.3, we can write an explicit form of the order of the second group of units of the ring of Gaussian integers modulo any non zero and non unit Gaussian integerβ given in the following theorem.

Theorem 3.1. Letβ=αn

r

t=1

pnt

t

j

s=1

πkss

be the decomposition ofβinto product of distinct

prime powers, with n,nt, ksare non negative integers for1≤t≤r and1≤s≤ j. Then

φ2G(β) =                r

t=1

fpt(nt)φ(φG(p nt

t )) j

s=1

φ(φG(πnss))

if n≤2 2

r

t=1

fpt(nt)φ(φG(p nt

t )) j

s=1

φ(φG(πkss))

if 3≤n≤4 2n−4

r

t=1

fpt(nt)φ(φG(p nt

t )) j

s=1

φ(φG(πkss))

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Proof. Using the multiplicative property ofφ2G,we can write

φ2G(β) =φ2G(αn)

r

t=1

φ2G(pntt) ! j

s=1

φ2G

πkss

!

.

Applying lemmas 3.1, 3.2 and 3.3, we deduce the explicit form stated forφ2G(β).

Example 3.1. Letβ=3.72(1+i)3(1+2i)2,thenφ2G(β) =φ2G(3.52(1+i)3(1+2i)2) =φ2G((1+

i)3)φ2G(3)φ2G(52)φ2G((1+2i)2). Hence, φG2(β) =2φ(9−1)(7−1)272(2)−4φ(49−1)φ(5(5−

1)) =36864.

4.

Properties for

φ

2G

(

β

)

It is well known thatφG(γ)|φG(β)wheneverγ|β. We will prove this property forφ2G.

Lemma 4.1. Letβ =αnand letγ|β, thenφ2G(γ)|φ2G(β).

Proof. Sinceγ|β, thenγ=αk, where 1≤k≤n.Now, consider all possible cases ofn.Refering

to lemma 3.1, we have the following three cases. Ifn≤2, thenφ2G(β) =φ2G(γ) =1.

Conse-quently,φ2G(γ)|φ2G(β).Ifn=3 orn=4, thenφ2G(β) =2 andφ2G(γ) =1 orφ2G(γ) =2.Hence, φ2G(γ)|φG2(β). If n≥5, then φG2( β) =2n−4. Since 1≤k≤n, then we have k<4 or k≥5.

Ifk<4, thenφ2G(γ) =1 orφ2G(γ) =2,thusφ2G(γ)|φ2G(β).Ifk≥5, thenφ2G(γ) =2k−4.Since

k≤n, then 2k−4|2n−4and we getφ2G(γ)|φ2G(β).

Lemma 4.2. Letβ =πnand letγ|β, thenφ2G(γ)|φ2G(β).

Proof. Sinceγ|β, thenγ=πk, where 1≤k≤n.Lemma 3.2 givesφ2G(β) =φ(qn−1(q−1))and φ2G(γ) =φ(qk−1(q−1)).Sincek≤n, thenqk−1(q−1)|qn−1(q−1).Andφ(m)|φ(n)whenever

m|n, we conclude thatφ2G(γ)|φ2G(β).

Lemma 4.3. Letβ =pnand letγ|β, thenφ2G(γ)|φ2G(β).

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k=1 ork>1.Ifk=1, thenφ2G(γ) =φ(φG(p))and in this caseφ2G(γ)|φ2G(β).Ifk>1, then φ2G(γ) = (p−1)2p2k−4φ(φG(p)). But p2k−4|p2n−4fork≤n, thenφ2G(γ)|φ2G(β).

Theorem 4.1. Ifγ|β, thenφ2G(γ)|φ2G(β).

Proof. Letβ =αn

r

t=1

pnt

t

j

s=1

πkss

be the decomposition ofβ into product of distinct prime

powers, with n,nt and ks are non negative integers for 1≤t ≤r and 1≤s≤ j. As γ|β, we

write γ =αn0

r

t=1

pn0t

t

j

s=1

πk 0 s

s

with 0≤n0≤n, 0≤nt0≤nt and 0≤k0s ≤ks. Since φ2G is

multiplicative, then

φ2G(β) =φ2G(αn)

r

t=1

φ2G(ptnt) ! j

s=1

φ2G

πkss

!

andφ2G(γ) =φ2G(αn 0

)

r

t=1

φ2G pn 0 t t j ∏

s=1

φ2G πk 0 s s

. Using lemmas 4.1, 4.2 and 4.3, we get

φ2G(αn0)|φG2(αn),φ2G(πk 0 s

s )|φ2G(π ks

s )andφ2G(p n0t

t )|φ2G(p nt

t ). Hence the result.

Note that for a non trivial ringR with identity, it is always true that |U(R)|<|R| and con-sequently |U2(R)|<|U(R)|. In the next proposition, we aim to find a least upper bound for

φ2G(β).

Proposition 4.1. φ2G(β)≤ φG(β)

2 ifβ 6=α.

Proof. We consider the three cases according to the three different types of the Gaussian primes, then we use the multiplicative property to complete the proof for any non zero, non unit Gauss-ian integerβ.

First, we consider the case whereβ =αn.

Ifn=2, thenφ2G(α2) =1. ButφG(α2) =2,Consequently,φ2G(α2) = φG(α

2)

2 .

Ifn=3, thenφ2G(α3) =2 andφG(α3) =4.Againφ2G(α3) =φG(α

3)

2 .

Ifn=4, thenφ2G(α4) =2 andφG(α4) =8.Hence,φ2G(α4) < φG(α

4)

2 .

If n≥5, then φ2G(αn) = 2n−4 and φG(αn) =2n−1. Hence, φG2( αn) < φG(α

n)

2 . Conse-quently,φ2G(αn) ≤ φG(α

n)

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Ifn=1, thenφ2G(p) =φ(φG(p)) =φ(p2−1). Sincep2−1 is even, thenφ(p2−1)≤ p

21

2 andφ2G(p) =φ(p2−1)≤ p

21

2 =

φG(p)

2 .

Ifn>1, thenφ2G( pn) = (p−1)2p2n−4φ(φG(p))<p2n−2φ(p2−1).Then φ2G( pn)< p

2n−2(p21)

2 =

φG(pn)

2 . Consequently,φ

2

G( pn)≤

φG(pn)

2 .

Now, consider the case whenβ =πn.

Thenφ2G(πn) =φ(qn−1(q−1)).Sinceqn−1(q−1)is even, thenφ(qn−1(q−1))≤q

n−1(q1)

2 =

φG(πn)

2 . Therefore,φ

2

G(πn)≤

φG(πn)

2 .

Finally, let β =αn

r

t=1

pnt

t

j

s=1

πkss

be the decomposition of β into product of distinct

prime powers, withn,nt,ks are non negative integers for 1≤t≤rand 1≤s≤ j.Then,

φ2G(β) =φ2G(αn)

r

t=1

φ2G(pntt) ! j

s=1

φ2G

πkss

!

.

Using the above results, we can write

φ2G(αn)

r

t=1

φ2G(pntt) ! j

s=1

φ2G

πkss

!

φG(αn)

r

t=1

φG(pntt)

j

s=1

φG

πkss

2i+j+1

<

φG(αn)

r

t=1

φG(pntt)

j

s=1

φG

πkss

2 .

Consequently,φ2G(β)≤φG(β)

2 .

5.

Conclusion

Using the decomposition of the generalized second group of units of a quotient ring of Gaussian integersR=Z[i]/ <β >,we were able to generalize the Euler Phi function. The new

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Conflict of Interests

The author declares that there is no conflict of interests.

REFERENCES

[1] J. T. Cross, The Euler’sφ-function in the Gaussian Integers, Amer. Math. Monthly 90 (1983), 518-528.

[2] A. N. El-Kassar and H. Y. Chehade, Generalized Group of Units, Mathematica Balkanica, New Series 20

(2006), 275-286.

[3] Haissam Y. Chehade and Iman M. AL Saleh, Trivial Generalized Group Of units Of The RingZ[i]/ <β >, J.

Math. Comput. Sci. 6 (2016), No. 2, 203-215.

[4] Manal A. Assaf, On the Equations Involving the Order of the Second Group of Units of the RingZn, A master

thesis submitted to Lebanese International University,2016.

[5] Niven, I., and Zuckerman. H.S. An Introduction to the theory of numbers, 5th edition John Wiley and Sons,

References

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