THE IMPACT OF THE MONOID HOMOMORPHISM ON THE STRUCTURE OF SKEW GENERALIZED POWER

© 2018 Pushpa Publishing House, Allahabad, India http://www.pphmj.com

http://dx.doi.org/10.17654/MS103071215

Volume 103, Number 7, 2018, Pages 1215-1227 ISSN: 0972-0871

Received: October 4, 2017; Accepted: January 10, 2018 2010 Mathematics Subject Classification: 16W55.

Keywords and phrases: monoid homomorphism, strictly ordered monoid, skew generalized power series ring.

THE IMPACT OF THE MONOID HOMOMORPHISM ON THE STRUCTURE OF SKEW GENERALIZED POWER

SERIES RINGS

Ahmad Faisol^1,2, Budi Surodjo¹ and Sri Wahyuni¹

1Department of Mathematics Universitas Gadjah Mada Indonesia

2Department of Mathematics Universitas Lampung Indonesia

Abstract

Let R be a ring, (S,≤) be a strictly ordered monoid and ω:S → ( )R

End be a monoid homomorphism. In this paper, we study the properties of monoid homomorphism ω and its impact on the structure of skew generalized power series ring R[[S,ω]]. We show that: if

( )¹ ~ω( )²,

ω then R₁[[S,ω^{( )1}]]~= R₂[[S,ω^{( )2} ]], and R₁⊕R₂[[S, ( ) ( )]]~ [[ , ^{( )}]] [[ , ^{( )2} ]].

1 2 2 1

1 ⊕ω = ω ⊕ ω

ω R S R S

1. Introduction

In 2007, Mazurek and Ziembowski [1] constructed a new ring which is the generalization of generalized power series rings (GPSR) R

[ ] [ ]

S that was

constructed by Ribenboim [2] by using a monoid homomorphism ω:S →

( )

R

End to change the convolution product on GPSR R

[ ] [ ]

S . Furthermore, this new ring is known as skew generalized power series ring (SGPSR) denoted by R

[ [

S, ω

] ]

or R

[ [

S, ω, ≤

] ]

. Now we will give the definition and some examples of SGPSR R

[ [

S, ω

] ]

.

Regarding ordered sets, ordered monoids, artinian and narrow set, we will follow the terminology used in [2-6]. Now, we recall the construction of SGPSR [1]. Let

(

S, ≤

)

be a strictly ordered monoid, R be a commutative ring with an identity element and ω:S → End

( )

R be a monoid homomorphism. For any s∈S let ω_s denote the image of s under ω, i.e.,

( )

s = ω_s. ω

Define R^S =

{

f | f :S → R

}

and R

[ [

S, ω

] ]

=

{

f ∈R^S|supp

( )

f is artinian and narrow where

}

, supp

( )

f =

{

s∈S| f

( )

s ≠ 0

}

.

Under pointwise addition and skew convolution, multiplication defined by

( )( ) ( ) ( ( ) )

( )

∑

( )

χ

∈

ω

=

g f y x

x

s

y g x f s

fg

, ,

, (1)

for all f, g∈R

[ [

S,ω

] ]

, where

(

f g

) { (

x y

)

supp

( )

f supp

( )

g xy s

}

s = ∈ × | =

χ , ,

is finite, R

[ [

S, ω

] ]

is a ring which is known as skew generalized power series ring (SGPSR).

Some special cases of SGPSR R

[ [

S, ω

] ]

are given by the following example.

Example 1.1. Let R be a ring, id_R be an identity map in End

( )

R , N₀ be a set of positive integers, Z be a set of integers and

(

S, ≤

)

be a strictly ordered monoid.

(1) If S = N₀ with usual addition, trivial order ≤ and ω_s =id_R, for all ,

S

s∈ then SGPSR R

[ [

S, ω

] ]

is polynomial ring R

[ ]

X .

(2) If S = Z with usual addition, trivial order ≤ and ω_s = id_R, for all ,

S

s∈ then SGPSR R

[ [

S, ω

] ]

is Laurent polynomial ring R

[

X, X⁻¹

]

. (3) If S = N₀ with usual addition, trivial order ≤ and ω₀ = σ, for some endomorphism ring σ∈End

( )

R , then SGPSR R

[ [

S,ω

] ]

is skew polynomial ring R

[

X;σ

]

.

(4) If S = N₀ with usual addition, usual order ≤ and ω_s = id_R, for all ,

S

s∈ then SGPSR R

[ [

S, ω

] ]

is power series ring R

[ ] [ ]

X .

(5) If S = Z with usual addition, usual order ≤ and ω_s =id_R, for all ,

S

s∈ then SGPSR R

[ [

S, ω

] ]

is Laurent series ring R

[[

X, X⁻¹

]]

.

(6) If S = N₀ with usual addition, usual order ≤ and ω₀ = σ, for some endomorphism ring σ∈End

( )

R , then SGPSR R

[ [

S, ω

] ]

is skew power series ring R

[ [

X;σ

] ]

.

(7) If ω_s = id_R, for all s∈S, then SGPSR R

[ [

S, ω

] ]

is generalized power series ring

[[

R^(S,≤)

]]

= R

[ ] [ ]

S .

2. Main Results

In this section, we give the definition and some properties of monoid homomorphism ω and its impact on the structure of SGPSR R

[ [

S, ω

] ]

. First, we give the definition of equivalency of two monoid homomorphism.

Definition 2.1. Let R₁ and R₂ be rings,

(

S,≤

)

be a strictly ordered monoid, and ω^{( )1} :S → End

( )

R₁ and ω^{( )2} :S → End

( )

R₂ be monoid homomorphisms. Then ω^{( )1} and ω^{( )2} are said to be equivalent if there exists an isomorphism ϕ:R₁ → R₂ such that ω^{( )}_s² = ϕω^{( )}_s¹ϕ⁻¹ for all s∈S. In this case, we write ω^{( )1} ~ ω^{( )2} .

Example 2.2. Let S = N₀, R₁ = Q×Q = Q² and R₂ = Z×Z = Z². With operation

(

x, y

) (

+ m, n

) (

= x +m, y +n

)

and

(

x, y

) (

m,n

) (

= xm, yn

)

,

R1 and R₂ become rings and S becomes a strictly ordered commutative monoid with pointwise addition and usual order. For any s∈S,

(

p, q

)

∈R₁ and

(

x, y

)

∈R₂, we define monoid homomorphism

( )¹ :S → End

( )

R₁ , ω

where ω^{( )}_s¹

(

p, q

) (

= 0, q

)

, and

( )² :S → End

( )

R₂ , ω

where ω^{( )}_s²

(

x, y

) (

= x,0

)

. Next, we define a map

2

:R1→ R ϕ

with ϕ

(

p, q

) (

= q, p

)

for all

(

p, q

)

∈R₁. Since for any

(

p, q

) (

, m,n

)

∈R₁, imply

( ) ( )

(

p q + m n

)

=ϕ

( (

p+m q+ n

) )

ϕ , , ,

(

q+n p +m

)

= ,

(

q, p

) (

+ n, m

)

=

( )

(

p, q

)

+ϕ

( (

m, n

) )

ϕ

= and

( ) ( )

(

p, q m,n

)

= ϕ

( (

pm, qn

) )

ϕ

(

qn, pm

)

=

(

q, p

) (

n, m

)

=

( )

(

p, q

) (

ϕ

(

m, n

) )

, ϕ

= ϕ is a ring homomorphism.

Furthermore, if ϕ

( (

p, q

) )

= ϕ

( (

m, n

) )

, then

(

q, p

) (

= n, m

)

, which is n

q = and p = m. In other words, we have

(

p, q

) (

= m,n

)

. Hence, ϕ is an injective homomorphism. For any

(

x, y

)

∈R₂, there exists

(

p, q

)

∈R₁ with

y

p= and q=x such that ϕ

( (

p,q

) ) (

= q, p

) (

= x,y

)

. Then, ϕ is a surjective homomorphism. In other words, ϕ:R₁ → R₂ is a ring isomorphism.

Moreover, since

( )_s²ϕ

( (

p,q

) )

= ω^{( )}_s²

(

ϕ

( (

p, q

) ) )

ω

( )

( (

q p

) )

s² ,

ω

=

(

q,0

)

=

( )

(

0, q

)

ϕ

=

(

ω^{( )}_s¹

( (

p, q

) ))

ϕ

=

( )_s¹

( (

p, q

) )

, ϕω

= ( )¹ ~ ω( )² .

ω

Based on Definition 2.1, the impact of equivalency of two monoid homomorphisms on the structure of SGPSR R

[ [

S,ω

] ]

is given by the following proposition.

Proposition 2.3. Let R₁ and R₂ be rings,

(

S, ≤

)

be a strictly ordered monoid, and ω^{( )1} :S → End

( )

R₁ and ω^{( )2} :S → End

( )

R₂ be monoid homomorphisms. If ω^{( )1} ~ ω^{( )2} , then R₁

[[

S, ω^{( )1}

]]

~= R₂

[[

S, ω^{( )2}

]]

.

Proof. Suppose ω^{( )1} ~ ω^{( )2} . Then by Definition 2.1, there exists an isomorphism ϕ:R₁→ R₂ such that ω^{( )}_s² = ϕω^{( )}_s¹ϕ⁻¹ for all s∈S. Next, we define a map

[[

, ^{( )}

]] [[

, ^{( )}

]]

,

: ₁ ω¹ → ₂ ω²

ψ R S R S

where ψ

( )

f = f = ϕD f for all f ∈R₁

[[

S, ω^{( )1}

]]

.

For all s∈S and f, g ∈R₁

[[

S,ω^{( )1}

]]

, we have

(

f + g

) ( )

s = ϕ

( (

f + g

) ( )

s

)

ϕD

( ) ( ) (

f s + g s

)

ϕ

=

( )

(

f s

)

+ϕ

(

g

( )

s

)

ϕ

=

(

ϕD f

) ( ) (

s + ϕDg

) ( )

s

= and

( )

(

ϕD fg

) ( )

s = ϕ

( ( ) ( )

fg s

)

( )

^{( )}

( ( ) )

_⎟^⎟

⎠

⎞

⎜⎜

⎝

⎛ ω

ϕ

=

∑

=xy s

x g y

x

f ¹

( ( )

^{( )}

( ( ) ))

∑

=

ω ϕ

=

xy s

x g y

x

f ¹

( )

( ) (

^{( )}

( ( ) ))

∑

=

ω ϕ ϕ

=

xy s

x g y

x

f ¹

( ) ( ) (

^{( )}

) ( ) ( )

∑

=

ω ϕ ϕ

=

xy s

x g y

x

f D ¹

D

( ) ( ) (

^{( )}

) ( ) ( )

∑

=

ϕ ω ϕ

=

xy s

x g y

x

f D

D ²

( ) ( )

^{( )}

( ( ( ) ) )

∑

=

ϕ ω ϕ

=

xy s

x g y

x

f ²

D

( ) ( )

^{( )}

( ( ) ( ) )

∑

=

ϕ ω ϕ

=

xy s

x g y

x

f D

D ²

( ) ( )

(

ϕD f ϕDg

) ( )

s .

=

Since supp

( )

f ⊆ supp

( )

f , f ∈R₂

[[

S, ω^{( )2}

]]

. Then, we have

(

f + g

)

= f + g

ψ

(

f + g

)

ϕ

= D

(

ϕD f

) (

+ ϕDg

)

= g f +

=

( )

f +ψ

( )

g ψ

= and

( )

fg = fg ψ

( )

fg ϕD

=

(

ϕD f

) (

ϕDg

)

= g

= f

( ) ( )

f ψ g , ψ

=

for all f, g∈R₁

[[

S, ω^{( )1}

]]

. In other words, the map ψ:R₁

[[

S, ω^{( )1}

]]

→

[[

^{( )2}

]]

2 S, ω

R is a ring homomorphism.

Now, we will show that ψ is injective. Let f ∈Ker

( )

ψ . Then

( )

= 0.

ψ f Then, for all s∈S, we have

(

ϕD f

) ( )

s =0

( )

s . In other words,

( )

(

f s

)

ϕ = 0. Since ϕ is a ring isomorphism, f

( )

s = 0, for all s∈S. Then

( )

ψ

Ker = 0, so ψ is injective.

Furthermore, we will show that ψ is surjective. For all g∈R₂

[[

S,ω^{( )2}

]]

, there exists h=ϕ⁻¹Dg∈R₁

[[

S,ω^{( )1}

]]

such that ψ

( )

h =h =ϕDh=ϕDϕ⁻¹g

.

=g Then ψ is surjective. So R₁

[[

S, ω^{( )1}

]]

~= R₂

[[

S,ω^{( )2}

]]

.

Now we will give the definition of direct sum of two monoid homomorphisms.

Definition 2.4. Let R₁ and R₂ be rings,

(

S,≤

)

be a strictly ordered monoid, and ω^{( )1} :S → End

( )

R₁ and ω^{( )2} :S → End

( )

R₂ be monoid homomorphisms. Then the direct sum of ω^{( )1} and ω^{( )2} is defined by

( )¹ ⊕ω( )² :S → End

(

R₁⊕R₂

)

, ω

where

(

ω^{( )1} ⊕ω^{( )2}

) (

_s r₁, r₂

) (

= ω^{( )}_s¹

( )

r₁ ,ω^{( )}_s²

( ))

r₂ ,

for all s∈S and

(

r₁, r₂

)

∈R₁⊕R₂.

Example 2.5. Let monoid S, rings R₁ and R₂,ω^{( )1} and ω^{( )2} be given as in Example 2.2. Then, we can define the direct sum of ω^{( )1} and ω^{( )2} by

( )¹ ⊕ω( )² :S → End

(

R₁⊕R₂

)

, ω

where

(

ω^{( )1} ⊕ω^{( )2}

) (

_s

(

p, q

) (

, x, y

) ) (

= ω^{( )}_s¹

( (

p, q

) )

, ω^{( )}_s²

( (

x, y

) ))

( ) ( )

(

0, q , x,0

)

,

=

for all s∈S and

( (

p, q

) (

, x, y

) )

∈R₁⊕R₂.

The following lemma shows that the direct sum ω^{( )1} ⊕ω^{( )2} that defined in Definition 2.4 is a monoid homomorphism.

Lemma 2.6. Let R₁ and R₂ be rings,

(

S,≤

)

be a strictly ordered monoid, and ω^{( )1} :S → End

( )

R₁ and ω^{( )2} :S → End

( )

R₂ be monoid homomorphisms. Then the direct sum

( )¹ ⊕ω( )² :S → End

(

R₁⊕R₂

)

ω

is a monoid homomorphism.

Proof. For any s,t∈S and

(

r₁, r₂

)

∈R₁⊕R₂, we have

(

ω^{( )1} ⊕ω^{( )2}

) (

_st r₁, r₂

) (

= ω^{( )}_st¹

( )

r₁ ,ω^{( )}_st²

( ))

r₂

((

ω^{( ) ( )}_s¹ω_t¹

) ( ) (

r₁ , ω^{( ) ( )}_s²ω_t²

) ( ))

r₂

=

(

ω^{( )}_s¹

(

ω^{( )}_t¹

( ))

r₁ ,ω^{( )}_s²

(

ω^{( )}_t²

( )))

r₂

=

(

ω^{( )1} ⊕ω^{( )2}

) (

_s ω_t^{( )1}

( )

r₁ ,ω^{( )}_t²

( ))

r₂

=

((

ω^{( )1} ⊕ω^{( )2}

) (

_s ω^{( )1} ⊕ω^{( )2}

) )(

_t r₁r₂

)

.

= Hence, we obtain

(

ω^{( )1} ⊕ω^{( )2}

)( ) (

st = ω^{( )1} ⊕ω^{( )2}

)( )(

s ω^{( )1} ⊕ω^{( )2}

)( )

t . So the direct sum ω^{( )1} ⊕ω^{( )2} is monoid homomorphism.

Now, based on Definition 2.4 and Lemma 2.6 we get the following proposition.

Proposition 2.7. Let R₁ and R₂ be rings,

(

S, ≤

)

be a strictly ordered monoid, and ω^{( )1} :S → End

( )

R₁ and ω^{( )2} :S → End

( )

R₂ be monoid homomorphisms. Then

(

R₁⊕R₂

) [[

S, ω^{( )1} ⊕ω^{( )2}

]]

~= R₁

[[

S, ω^{( )1}

]]

⊕R₂

[[

S,ω^{( )2}

]]

.

Proof. Let i₁: R₁ → R₁⊕R₂ and i₂ :R₂ → R₁⊕R₂ be natural injections, and let p₁:R₁⊕R₂ → R₁ and p₂ :R₁⊕R₂ → R₂ be natural projections. Then we have

( )_s¹ = p₁

(

ω^{( )1} ⊕ω^{( )2}

)

_si₁ ω

and

( )_s² = p₂

(

ω^{( )1} ⊕ω^{( )2}

)

_si₂, ω

as seen in the following diagram:

Then we obtain

( )_s¹ p₁ = p₁

(

ω^{( )1} ⊕ω^{( )2}

)

_si₁p₁ ω

(

^{( )1 ( )2}

)

₁

1 sidR

p ω ⊕ω

=

(

^{( ) ( )}

)

_s p₁ ω¹ ⊕ω²

= and

( )_s² p₂ = p₂

(

ω^{( )1} ⊕ω^{( )2}

)

_si₂p₂ ω

(

^{( )1 ( )2}

)

₂

2 sidR

p ω ⊕ω

=

(

^{( )1 ( )2}

)

.

2 s

p ω ⊕ω

=

Now, for any f ∈

(

R₁⊕R₂

) [[

S, ω^{( )1} ⊕ω^{( )2}

]]

, we define a map

(

_{1 2}

) [[

, ^{( )1 ( )2}

]]

₁

[[

, ^{( )1}

]]

₂

[[

, ^{( )2}

]]

: ⊕ ω ⊕ω → ω ⊕ ω

ψ R R S R S R S

by ψ

( ) (

f = f₁, f₂

)

, where f₁ = p₁D f and f₂ = p₂ D f.

For ,i=1,2 we will show p_iD

(

f +g

) (

= p_iDf

) (

+ p_iDg

)

and p_iD

( )

fg

(

p_i D f

) (

p_i Dg

)

.

= For any s∈S, f, g ∈

(

R₁⊕R₂

) [[

S, ω^{( )1} ⊕ω^{( )2}

]]

and ,

2 ,

=1

i we have

( )

(

p_i D f + g

) ( )

s = p_i

( (

f + g

) ( )

s

) ( ) ( ) (

f s g s

)

p_i +

=

( )

(

f s

)

p

(

g

( )

s

)

p_i + _i

=

(

p_i D f

) ( ) (

s + p_i Dg

) ( )

s

= and

( )

(

p_i D fg

) ( )

s = p_i

( ( ) ( )

fg s

)

( ) (

^{( ) ( )}

) ( ( ) )

_⎟^⎟

⎠

⎞

⎜⎜

⎝

⎛ ω ⊕ω

=

∑

=xy s

i f x s g y

p ^{1 2}

( ) (

^{( ) ( )}

) ( ( ) )

∑

=

ω

⊕ ω

=

xy s

i s

if x p g y

p ^{1 2}

( )

^{( )}

( ( ) )

∑

=

ω

=

xy s

i s

if x p g y

p ¹

( ) ( )

^{( )}

( ( ) ( ) )

∑

=

ω

=

xy s

i s

i f x p g y

p D ¹ D

( ) ( )

(

p_i D f p_i Dg

) ( )

s .

=

Since for any f, g∈

(

R₁⊕R₂

) [[

S,ω^{( )1} ⊕ω^{( )2}

]]

, we have

(

f + g

) ((

= f + g

) (

₁, f + g

) )

₂

ψ

( ) ( )

(

p f + g p f + g

)

= ₁D , ₂D

( ) ( ) ( ) ( )

(

p₁D f + p₁Dg , p₂D f + p₂ Dg

)

=

(

f₁+ g₁, f₂ +g₂

)

=

(

f₁, f₂

) (

+ g₁, g₂

)

=

( )

f +ψ

( )

g ψ

=

and

( ) (( ) ( ) )

fg = fg ₁, fg ₂ ψ

( ) ( )

(

p₁D fg , p₂ D fg

)

=

( ) ( ) ( ) ( )

(

p₁D f p₁Dg , p₂ D f p₂Dg

)

=

(

f₁g₁, f₂g₂

)

=

(

f₁, f₂

) (

g₁, g₂

)

=

( ) ( )

f ψ g , ψ

= ψ is a ring homomorphism.

Now, we will show ψ is injective. Let f ∈Ker

( )

ψ . Then we will show .

= 0

f Since f ∈Ker

( ) ( ) (

ψ , ψ f = 0, 0

)

. So, for any s∈S and i =1, 2, we have

(

p_i D f

)( )

s = 0

( )

s . In other words, p_i

(

f

( )

s

)

=0. Since p_i is a natural projection, f

( )

s = 0 for all s∈S. So Ker

( )

ψ =0 or ψ is injective.

Furthermore, we will show ψ is surjective. For all

(

f₁, f₂

)

∈R₁

[[

S,ω^{( )1}

]]

[[

, ^{( )2}

]]

,

2 ω

⊕R S there exists

[[

^{( ) ( )}

]]

∑

= ∈ ⊕ ω ⊕ω

= ²

1

2 1 2

1 ,

k ik fk R R S

f D

such that ψ

( ) (

f = f₁, f₂

)

.So, ψ is surjective. Then, ψ is a ring isomorphism.

So R₁⊕R₂

[[

S, ω^{( )1} ⊕ω^{( )2}

]]

~= R₁

[[

S, ω^{( )1}

]]

⊕R₂

[[

S,ω^{( )2}

]]

. References

[1] R. Mazurek and M. Ziembowski, Uniserial rings of skew generalized power series, J. Algebra 318 (2007), 737-764.

[2] P. Ribenboim, Generalized power series rings, Lattice, Semigroups and Universal Algebra, Plenum Press, New York, 1990, pp. 271-277.

[3] G. A. Elliott and P. Ribenboim, Fields of generalized power series, Arch. Math.

54 (1990), 365-371.

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Ahmad Faisol: ahmadfaisol@mail.ugm.ac.id; ahmadfaisol@fmipa.unila.ac.id Budi Surodjo: surodjo_b@ugm.ac.id

Sri Wahyuni: s_wahyuni@ugm.ac.id