Power Module

Fang Gang**[1]**

**Abstract:
** In this paper, we discuss the upgrade
problem of module, and introduce the concepts of the power module, regular
power module and uniform power module. We give some results of them

**Key
words:** power group; power module; regular
power module; uniform power module

1. Introduction

The
notion of the hypergroup was first introduced by LI Hong-xing and WANG Pei-zhuang in 1985(LI,
DUAN & WANG, 1985). Afterward, in 1988, LI Hong-xing and WANG
Pei-zhuang emphasized on the upgrade problem for algebraic group, and first introduced
the notion of power group (LI & WANG, 1988). In 1990, ZHONG Yu-bin investiaged further the structures of hypergroups(ZHONG, 1990). In 1988, LI Hong-xing established the HX
ring in hypergroups (power groups) (LI, 1988). Yao Bing-xue and LI Hong-xing introduced the concept of power ring and
improved some results of HX rings (YAO & LI, 2000). ZHANG Zhen-liang described
the normal power rings and uniform power rings in 2001(ZHANG, 2001). Nowdays, it has been seen that the upgrade of all kinds of structures such as algebraic structures,
ordered strucutres, topological structures, measurable structures is important on
the development of fuzzy mathematics.

In
this paper, we shall extend hypergroup and power ring to module theory by
introducing the notions of so called power module, regular power module and
uniform power module in a R-module.

Through
the paper, we always assume that the ring *R* is a commutative with
identity. By a *left R-module M*, we
shall mean* *an abelian group
(*M*, *+*) together with a left action *R
M M*, described by (*r*, *x*)* _{}rx*, such that for all

*r*,

*s*in

*R*, and all

*x*,

*y*in

*M*, we have

(1) *r*(*x+y*)* = rx+ry* ;

(2)
(*r+s*) *x = rx+sx *;

(3)
(*rs*) *x = r *(*sx*) ;

(4)
1*x = x*,* *where 1* *is
multiplicative identity element of *R.*

Suppose
*M* is a left *R*-module and *N* is a subgroup
of *M*. Then *N * is a

*submodule* (or *R-submodule*)
if, for any *n *in *N *and any *r* in *R*, the product *rn* is in *N*.

In
the following, we shall introduce the power sets into module. For the more details for Module theory we refer the reader to [8, 9].

Let* _{}*(

*M*)

*=*{

*A | A*Í

*M*} and

*(*

_{}*M*)

*=*(

_{}*M*)

*-*{

*f*}. For every

*A*,

*B*Î

*(*

_{}*M*) and Î

*R*, the sum of

*A*and

*B*is defined by

*A+B =*{ *a+b *| *a*Î*A*, *b*Î*B*},
(*)

and
the product of *B *and number
(Î*R*) is defined by

*B =*{*b | b*Î*B *}.* * (**)

Clearly,
we have

**Proposition1.1 **Let *A*,
*B*, *C** _{}*(

*M*)

*and , Î*

*R.*Then we have

(1) *A+B=B+A*

(2)
(*A+B*)*+C=A+*(*B+C*)

(3)
(*˦*) *C= *(*C*)

(4) (*B+C*)*=B+C*

(5)
(*+*) *A *Í* A+A*

(6)
1*A*=*A*, (* *where 1* *is multiplicative identity element of *R*)

(7)
If *A *Í *B*,
then *A+C *Í* B+C*, and *A *Í *B*.

**Definition 1.1 **Let *M *be a non-empty
subset of * _{}*(

*M*). If

*M*

*forms a left*

*R-*module

*M*under the operation (*) and (**), then

*M*

*is called a*

*power module*(

*or*

*R-*

*power module*) on

*M*, whose

*null element*is denoted by

*Q*and the

*nagative element*of

*A*is denoted by -

*A*. The set

*{*

_{}=*x*| -

*x*Î

*A*

*x*Î

*M*}

*is called the*

*inverse*of

*A*.

Clearlyfor every Î*R *we have *Q=Q*.

**Example 1** Let *S* be a submodule of *M*. Then* *{{*x*} | *x*Î*S*} is an *R-*power module on *M*.

**Example 2 **Let *S*
be a submodule of *M*. Then the factor
(quotient) module* M/S=*{*x+S *|*
x*Î*M*} is an *R*-power module on *M*.

**Example 3** Let *R*
be a real field and *A=*{*X | **f** X *Í *Z*,
where* Z *is the set of integers}. Then
*R *is a left* R-*module, and *A* is a hypergroup relative to the
operation (*). Let 0.7Î*R* and {1, 2}Î*A* we have that 0.7{1, 2}*=*{0.7, 1.4}* _{}A*, hence

*A*is not a

*R*-power module of

*R*. Thus we can see that not all hypergroups

*are power modules.*

**Definition 1.2 **Let *M** *be a power module of *M*.Then *M** *is a *regular **power module* of *M** *, if 0Î*Q *(0 is the
zero element of* M *), and then *M** *is an *uniform **power module* of *M* if -*A= _{} *for every

*A*Î

*M*.

**Definition 1.3 **Let *M *be an *R-**power module* on *M*. For every *A*Î*M*, the set * _{}=*{

*a*|

*a*Î

*A*, -

*a*Î-

*A*} is called the

*kernel*of

*A*.

2. Power Module

**Theorem 2****.1 **** **Let *M *be a power module of *M*. Then we have

(1) For
every *A*Î*M*, we have | *A *|=| *Q *|;

(2) For
all *A*, *B*Î*M*if *AB**f* then | *A *|=|
*AB *|;

(3) For
all *A*, *B*Î*M*if* A *Í* B *then* *-*B
*Í -*A*.

**Proof **(1) Since *Q*
is zero element of *M*, so for every *a*Î*M* we have *a+Q *Í *A+Q
=A*, and hence |*Q*|=| *a+Q | * | *A *|.

Conversely,
since *-A+A=Q*then for every* b**-A* we have that *b+A
*Í *-A+A=Q*
and* | A |=| b+A | | Q*|. Consequencely,
*| A |=| B |=| Q |.*

2Since *AB**f* then we have | *AB *| | *A *|. Moreover, if
*z*Î*AB* then we have *z*Î*A* and *z*Î*B*hence
we see that* z+Q *Í* A+Q=A* and *z+Q *Í* B+Q=B*. Thus we obtain *z+Q *Í *AB*, so
| *A *| = | *Q *|=| *z+Q *| | *AB *|. Consequently, we have | *A *|=| *AB *|.

3By *A
*Í* B* we have *-A-B+A *Í* B-A-B*, so *CB *Í* -A*.

**Theorem 2.2 **Let *M *be a power
module of *M*. If 0Î*A*Î*M*then
-*A *Í *Q
*Í *A*.

**Proof** Since
0*A*then *Q=*0*+Q *Í* A+Q=A*. On the other hand, we have that -*A* *=*0*+*(-*A*)* *Í* A+*(-*A*)*=Q*.
Thus we have *CA *Í* Q *Í *A*.

**Corollary 2.1 **Let *M *be a power
module of *M*. If 0Î*A*Î*M** *and |*M** *| is finite, then
-*A=Q=A*.

**Proof** Since
|*M** *| is finite, so
are *A* and *Q*. By** **Theorem 2.2 we see C*A *Í* Q *Í* A*, hence* |*-*A | | Q | | A |*.
Moreover, we have *| A |=| *-*A |*, thus *|A|=| *-*A |=| Q |*.
Consequently, we obtain -*A=Q=A*.

**Theorem 2.3 **Let *M *be a power
module of *M* and the zero element *Q *be a submodule of *M*. Then _{} Í _{}.

**Proof**Let *b*Î* _{}*, then

*-b*Î

*A*, and for every

*c*Î

*we have*

_{}*-b+ c*Î

*A- A= Q*. Since

*Q*is a submodule of

*M*, then there exists a nagative element

*t*Î

*Q*such that

*c-b*+

*t=*0 , namely

*t=b-c*, whence

*b=c+ t*Î

*, and consequently, we have*

_{}+Q =_{}_{}Í

_{}.

**Theorem 2.4 **Let *M *be a power
module of *M*, then *M *is a regular power module Û_{}*f*, for every *A**M**.*

**Proof**Ü* * If _{}*f*, then there exist *a*Î*A* and -*a*Î-*A*, so
we have 0*=a*-*a*Î*A*-*A=Q*.
Consequently, *M *is a regular power module.

Þ For
every *A*Î* we have **A*-*A=Q*.
Since *M** *is a regular
power module, then 0Î*Q* so there exist *a*Î*A* and *b*Î-*A*,
such that *a+ b= *0. Hence *-a=
b*Î-*A*,* *and whence *a*Î_{}*f*.

**Theorem 2.5** Let *f*
: *L*_{1}*L*_{2 }be an *R-*morphism and *L*_{11 }be a *R-*power module on *L*_{1}, then *L*_{22}*=*{ *f*(*A*) | *A*Î*L*_{11}} is an *R-*power module on *L*_{2} and *L*_{11}~*L*_{22}.

**Proof**It is easy to see from [7] that *L*_{22}*=*{ *f*(*A*)* *| *A*Î*L*_{11}} forms an additive power group with null element *f *(*Q*)
and for all *A*,* B*Î*L*_{11}, we have* f*(*A*)*+f*(*B*) *=
f *(*A+B *, -*f*(*A*)* = f*(-*A*). Moreover, let Î*R* ,* A*Î*L*_{11 }and *t*Î*f(A) *then we have that *t*_{1}Î*A _{ }*and

*t=f*(

*t*

_{1})

*=f*(

*t*

_{1})Î

*f*(

*A*), hence

*f*(

*A*) Í

*f*(

*A*). For the converse inclusion, we let

*h*Î

*f*(

*A*),

*then*

*h*

_{1}Î

*A*and

*h=f*(

*h*

_{1})

*=f*(

*h*

_{1})Î

*f*(

*A*). Hence

*f*(

*A*) Í

*f*(

*A*). Thus we obtain

*f*(

*A*

_{1})

*=f*(

*A*

_{1}). Consequently,

*L*

_{22}

*=*{

*f*(

*A*)

*|*

*A*Î

*L*

_{11 }} is an

*R-*power module on

*L*

_{2}and

*g: L*

_{11}

*L*

_{22}defined by

*g*(

*A*)

*= f*(

*A*) is an

*R-*epimorphism.

**Theorem 2.6** Let *f*
: *L*_{1}*L*_{2 }be an *R-*epimorphism and *L*_{22 }be a *R-*power module on *L*_{2}, then *f* ^{-1}(*L*_{22})*=*{ *f *^{-1}(*A*) | *A*Î* L*_{22}} is an *R-*power module on *L*_{1} and *f* ^{-1}(*L*_{22})*L*_{22}.

**Proof ** It is
clear from [7] that *f* ^{-1}(*L*_{22}) forms an additive power group. For *A** **L*_{22} and* *Î*R* ,we have *f ^{ -}*

^{1}(

*A*) =

*f*

^{ -}^{1}

*(*

*A*). Hence

*f*

^{-1}(

*L*

_{22})

*is an*

*R-*power module on

*L*

_{1}and

*g: f*

^{-1}(

*L*

_{22})

*L*

_{22}

*defined by*

*g*(

*f*

^{ -}^{1}(

*A*))

*=A*is an

*R-*isomorphism.

**Theorem 2.7 **Let *M *be a power
module of *M* and *Q* be a subgroup of *M*, then
*M*^{*}*=*{*N *|*
N*Î*M** *} is a left* R-*module.

**Proof** By
Theorem 2.1 in [4] and Theorem 2.2, we have that* **M** ^{*}* is a
subgroup of

*M*.

For every* N*Î*M** ^{* }*and

*Î*

*R*, by Definition 1.1 we have

*N*Î

*M*, and so

*N*Î

*M*

*. Thus*

^{*}*M*

*is a left*

^{*}*R-*module.

3. Regular power module and uniform power module

**Theorem 3.1 ** Let *M *be a regular power module of *M*.
If *A*Î*M* and *a*Î*A* , then *a*Î* _{}*Û

*A=a+Q*.

**Proof****Ü Since 0Î***Q*, then *a=a+*0Î*A+Q=A* and *Q=A*-*A=
*(*a+Q*) *-A =a+*(*Q*-*A*)*=a-A*.
Moreover, by 0Î*Q* and *a*Î*A*, we obtain 0Î* a-A*, then there
exists *b*Î-*A*
such that 0*=a*+*b*, namely *-a=b*Î-*A*. Thus
aÎ_{}.

Þ We can see clearly that *a*+*Q *Í* A*+*Q=A*.
By the definition 1.3 and *a*Î* _{}*,

**we have that**

*a*

*A*and

*-a*Î

*-A*. Let

*b*Î

*A*, then we have that

*b=*0

*+b=*(

*a-a*)

*+b=a+*(

*b-a*) Î

*a+A+*(-

*A*)

*=a+Q*. Hence we have

*A*Í

*a+Q*and whence

*A=a+Q*.

**Theorem 3.2**
Let *M *be a regular power module of *M*. If *A*Î*M** *and *a*Î*A*, then *a*Î* _{}*Û

*+*

_{}=a_{}.

**Proof****Ü It follows that 0Î*** _{}* and
hence

*a=a+*0Î

*a+*

_{}=_{}.Þ For evey *x*Î* _{},*
then

*x*Î

*A*and

*-x*Î

*-A*, and by Theorem 3.1 we have

*x*Î

*A=a+Q*. Hence there exists

*b*Î

*Q*such that

*x=a+b*. Since

*b=x-a*and so -

*b=a*-

*x*Î (-

*A*)

*+A =Q*. We thus have

*b*Î

*, and*

_{}*x=a+b*Î

*a+*, namely

_{}_{}Í

*a+*. For the converse inclusion we let

_{}

*y*Î

*a+*Then by Theorem 3.1, there exists

_{}.*b*Î

*Í*

_{}*Q*such that

*y=a+b*Î

*a+Q=A*. Since

*b*Î

*and*

_{}*a*Î

*, then*

_{}*-b*Î

*-Q=Q*and

*-a*Î

*-A*, hence

*-y=*(

*-b*)

*+*(

*-*a)Î

*Q-A=-A*. It follows that

*y*Î

*, and so*

_{}*a+*Í

_{}*. Consequently*

_{}*.*

_{}=a+_{}**Theorem 3.3** Let *M *be a regular power module of *M*. Then *M *is an uniform
power module of *M *Û * _{}=A*,

*for every*

*A*Î

*M*.

**Proof**Þ Since *M *is a regular power module of *M*,
for every *A*Î*M *and every *a*Î*A* we have *-a*Î* _{}=-A*,
namely

*a*Î

*, hence*

_{}*A*Í

*. Moreover, it is clear to see that*

_{}*Í*

_{}*A.*Thus we have

*.*

_{}=AÜ Since *A*Î*M* and * _{}=A*, then for every

*a*Î

*A*we have

*a*Î

*, and by theorem 3.1, we have*

_{}*a+Q=A*and -(

*a+Q*)

*=*-

*A*, thus -(

*a+Q*)

*=*-

*a+Q = -A*.

Now,
we shall verify that * _{}=*-

*A*. Let

*b*Î

*then -*

_{}*b*Î

*A=a+Q*. Hence there exists

*s*Î

*Q*such that -

*b=a+s*. Since for every

*A*Î

*, we see*

*. Then*

_{}= A*and so there exists*

_{}=Q*-*

*s*Î

*Q*such that

*b= -a-s*Î-

*a+Q =*-

*A*. Thus

*Í*

_{}*-A*. On the other hand, if

*b*Î-

*A*, by -

*A=*-

*a+Q*we have

*b*Î-

*a+Q*, and hence there exists

*t*Î

*Q*such that

*b=t*-

*a*. Similarly, there exists

*-*

*t*Î

*Q*such that -

*b=a*-

*t*Î

*a+Q =A*. Hence we have

*b*Î

*, and so -*

_{}*A*Í

*. Thus*

_{}*-*

_{}=*A*, and consequently,

*M*is an uniform power module of

*M*.

**Theorem 3.4 **Let *M *be an uniform power module of *M*. Then *M*^{*}*=*{*A
*|* A*Î*M** *} is a submodule of *M*.

**Proof** By
Definition 1.2 and Theorem 2.1
in [4], we have that *M*^{*}* *is an additive subgroup of *M*. For every Î*R* and *a*Î* ^{*}*,
there exists

*A*Î

*M*such that

*a*Î

*A*,

*so*

*a*Î

*A*Í

*A*. Hence

*a*Î

*M*

*. Thus*

^{*}*M*

*={*

^{*}*A | A*Î

*M*

*} is a submodule of*

*M*.

**Theorem 3.5** * *is
an uniform power module of *M* Û *Q* is a
submodule of *M*.

**Proof**Þ By Definition 1.2 and Theorem 3.1 in [4], we see that *Q *is an additive subgroup of *M*.
If* *Î*R* then* Q=Q *Í* M*, so that *Q* is a submodule of *M*.

Ü Since *Q* is a submodule of *M*, then
we have that *Q *is an additive
subgroup of *M*. For every *A*Î*M*, we obtain by Theorem 2.2 in
[4] that -*A= _{}*. Thus

*M*

*is an uniform power module of*

*M*.

**Theorem 3.6 **(Structure theorem 1) Let *M *be a regular power
module of *M*. Then *M *={ *a+Q *| *a*Î* _{} *Í

*M*

*}, where*

^{** }*M*

^{**}*=*{

*Î*

_{}| A*M*

*}.*

**Proof** By
Definition 1.3, if *a*Î* _{}*
then we have that

*a+Q*Í

*A+Q=A*. On the other hand, for

*b*Î

*A*, since

*a*Î

*, we have that*

_{}*b=*0

*+b=*(

*a-a*)

*+b=a+*(

*b-a*) Î

*a-A+A=a+Q*, and hence

*A*Í

*a+Q*. Consequently,

*A=a+Q,*and then

*M*

*=*{

*a+Q*|

*a*Î

*Í*

_{}*M*

*}.*

^{** }**Corollary 3.1 **(Structure
theorem 2) Let *M** *be an uniform power module of *M*. Then* **M *={ *a+Q*
|* a*Î*A *Í *M** ^{* }*}, where

*M*

^{*}*=*{

*A*|

*A*Î

*M*

*}.*

**Proof** By
Theorem 3.5, we see that *Q* is a submodule of *M*, hence 0Î*Q*. Namely, *M* is a regular power
module of *M*, and by Theorem 3.3, we have that * _{}=A*. Consequently, we
have by Theorem 3.6 that

*M*

*=*{

*a+Q*|

*a*Î

*A*Í

*M*

*}.*

^{*}References

[1] LI
Hong-xing, DUAN Qin-zhi, WANG Pei-zhuang. (1985). Hypergroups[J]. *BUSERAL*, 23: 22-29.

[2] LI
Hong-xing. (1988). HX Rings[J]. *BUSERAL*,
34:3-8

[3] LI
Hong-xing. (1991). HX Rings[J]. *Chinese
Quarterly Journal of Mathematics*, 6(1): l5-20.

[4] ZHONG
Yu-bin. (1990). The Structure and Relationship on Hypergroup[J]. *Chinese Quarterly Journal of Mathematics*,
5(4): 102-106.

[5] YAO
Bing-xue, LI Hong-xing. (2000). Power Ring[J]. *Fuzzy Systems and Mathematics*, 14(2): 15-20.

[6] ZHANG
Zhen-liang. (2001). Normal power ring and uniform power ring[J]. *Pure and Applied Mathematics*, 17(1): 6-13.

[7] LI Hongxing,WANG
Pei-zhuang. (1988). The Power Group[J]. *Mathematics
Applicate*, 1988,1(1): 1-4.

[8]
Anderson F W, Fuller K R. (1992). *Rings
and Categories of Modules*[M]. New York Heidelberg
Berlin Spinger-Verlag.

[9] T.S. Blyth.
*Module theory*[M]. (1990). Oxford University
Press, Oxford.

[1] Vice professor, School of Computer Science, Guangdong Normal Polytechnic University,
Guangzhou, 510665, China.

* Received 5 February 2009; accepted 25 April 2009

DOI: http://dx.doi.org/10.3968%2Fg795

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