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International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2015): 78.96 | Impact Factor (2015): 6.391
Volume 6 Issue 7, July 2017
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
- Semiprime Submodules
Nuhad Salim Al-Mothafar1, Iman A. Athab2
Department of Mathematics, College of Science, Baghdad University, Baghdad, Iraq
Abstract: Let be a commutative ring with identity and be a proper submodule is called semiprime if whenever ;
, , +, implies that . In this paper we say that is -semiprime if whenever +(); , , +,
implies that . We prove some result of this type of submodules.
Keywords: prime submodule, semiprimesubmodule, nearly semi prime submodule, weekly semi prime submodule
1. Introduction
Let be a commutative ring with identity and is an R-
module, Danus [2] was named semiprime submodules that
they are generalized of semiprime ideals, which get a big
interest at last years, many students search are published
about semiprime submodules by many people who care with
the subject of commutative algebra and some of them are J.
Dauns, R. L. Mcsland, C. p. Lu, P. F. Smith and M. E.
Moore. The definition comes in [2] as follows: we say that a
proper submodule of an R-module is semiprime
submodule if whenever ; , and +
implies that . In [1], the most important result was
proved which says that a proper submodule of an R-
module is semi prime if whenever 2 ; ,
implies that . If is a proper submodule of an R-
module M, then the following statements are equivalent. is
semiprime submodule of , then [:] is semiprime ideal
of , for where := { : }, [1], then [:]
is semiprime ideal of , for all . In this paper we
introduce a new class of submodules which is called
={: is a maximal submodule of }-semiprime
submodule, for short we use -semiprime submodule, a
proper submodule of an R-module is said to be -
semiprime submodule of if whenever +
(); , and + implies that . We
prove important result that is called -semiprime if
whenever 2 +(); , implies that
, and we prove many new result. In this paper we
give the following characterization, ifis a submodule of an
R-mdoule such that = 0 , then the following
statement are equivalent:
1) is-semiprimesubmodule of .
2) is semiprime submodule of .
3) [:]is semiprime ideal of for each submodule of
containg properly.
4) [:]is semiprime ideal of , for all and .
Also we introduce another characterization for -semiprime
submoduke of an R-module M and some properties of this
class of submodules.
2. -semiprime submodules
Recall that a proper submodule of an R-module is
called semiprime submodule if and whenever
; , and + such that
implies that , [2].
We introduce the following definition:
Definition (2.1)
A proper submodule of an R-module is is said to be -
semiprime submodule of if whenever +
(); , and + implies that
Remarks and examples (2.2)
1) Every -semiprime submodule is semiprime. But the
converse is not truein general for example:
Let =2 4be a module over and
=(0,0), (1,0) is a submodule of . Now +=
{(0,0),(1,0),(0,2),(1,2)}, then 9(0,2) +() but
3(0,2) which means that is not -semiprime while
is semiprime submodule.
2) Let be a prime number the module ∞ over has no
-semiprime submodule since ∞ as Z-module has no
semiprimesubmodule,[2]. Therefore by the previous
remark ∞ over has no -semiprime submodule
3) Every maximal submodule of an R-module is an -
semiprime submodule.
Proof:
Let be a maximal submodule of an R-module , then
[:] is a prime ideal of ,[5]. Now, for , and
+, suppose that +(), hence .
Since N is maximal submodule, then is prime
submodule,.[5]. Therefore either or[:], in
each case we have . This implies that is -
semiprime submodule of .
In particular the module over , the only -semiprime
submodule is 0.
4. Let be an R-module. () is a submodule of
define as follows:
() ={: is a prime submodule of },[3].
If =(), then every submodule of is -
semiprime.
Proof:
Let be a prime submodule of an R-module , then [:]
is a prime ideal of ,.[5]. For , and +,
suppose that +(), hence , but is a
prime submodule of , therefore either or
[:], this implies that .
The following proposition gives an important
characterization for -semiprime submodule.
Paper ID: ART20175463
DOI: 10.21275/ART20175463
1051
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2015): 78.96 | Impact Factor (2015): 6.391
Volume 6 Issue 7, July 2017
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
Proposition (2.3):
A proper submodule of an -modue is -semiprime if
and only if whenever 2 +,for , ,
then .
Proof
:
Since is -semiprimesubmodule, then by definition (2.1),
we prove this direction. For the converse suppose that for
, and +, suppose that +().
Now 22 +(), by assumption this implies
that 1 . After a finite number of steps, we get that
, this implies that is -seniprime submodule of .
From the definition (2.1) we can define -semiprime ideal
of a ring as a proper ideal satisfy that whenever +
() for and +, then .
Proposition (2.4):
A proper ideal of a ring is -semiprime if and only if
+() = .
Proof:
is-semiprime ideal of , let +(); ,
+ such that +(). Since thus and therefore
+() = .For the converse, it is clear that from the
properties of the radical of an ideal.
Consider the following proposition.
Proposition (2.5):
If is -semiprime submodule of an R-module , then
[:] is -semiprimeideal of .
Proof:
It enough to show that :+()[:], let
:+(), then + such that
:+(), therefore =+, for some [:]
and (). Now for all ,= + +
(), but is is -semiprime submodule, then so
that [:], this complete the proof.
The converse of the previous proposition not true in general
as the following example show:
= as Z-module, if =< 4,0>, then := 0
is -semiprimeideal, but not-semiprimesubmodule of .
Proposition (2.6):
Let be a submodule of an -module , then the following
statement are equivalent:
1. [+:]is-semiprimeideal of for all submodule
of containing+() properly.
2.[+: ()]is-semiprimeideal of for all and
+().
Proof
1)2) We have to prove that +:+()
[+:] for and +(). Let
+:+(), thus + such that
+:+(), + ++=
, by (1), we get that +:+() = [+
:]. Therefore [+:] and hence +
(), this implies that [+: ()] , then
+:+() = [+:] for and
+()
2)1)For a submodule of containning +
properly, we want to show that +:+=
+:, let +:+(), then
+ such that +:+() thus
=+ for some [+:] and (). Now
for and +(), from (2)
+: ()+() = +: (). It is clear that
[+:], hence [+:] and
therefore this complete the proof.
We can prove the following proposition:
Proposition (2.7):
If is an -semiprime submodule of an R-module M, then
the ideal [+:] is -semiprime in for all
submodules of containg + properly.
Proof:
let +:+(), then + such that
+:+(), thus =+ for some
[::] and (), then for all , =
+ +(), but is is -semiprime submodule, so
that and hence :[+:],
therefore +:+() = [+:] this
means that [++] is -semiprime ideal of .
From proposition (2.6) and proposition (2.7) we can prove
the following corollary.
Corollary (2.8):
If a proper submodule of an R-module M is -semiprime,
then the ideal [+:] is -semiprime for all
and +.
Now we are ready to prove the following characterization.
Proposition (2.9):
Let be an R-module such that = 0 and be a proper
submodule of , then the following statement equivalent:
1.) -semiprimesubmule of .
2.) is semiprime submodule of .
3.) [:]is semiprime ideal of , for each submodule of
containg N properly.
4.) [:]is semiprime ideal of , for all and
Proof:
(2)3(4)by (proposition (1.4),[1])
(2) (1) Suppose that 2 +() for and
. Thus 2 , but is semiprime submodule of
, therefore either or [:] in each case we
conclude that and hence is -semiprimesubmule
of .
Recall that a submodule of an R-module is called
injective envelope of in , denoted by () and define
as follows:
()= {=: , such that ,
+}. It is clear that , [4].
Paper ID: ART20175463
DOI: 10.21275/ART20175463
1052
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2015): 78.96 | Impact Factor (2015): 6.391
Volume 6 Issue 7, July 2017
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
Proposition (2.10):
Let be a proper submodule of an R-module , then -
semiprime if and only if =+().
Proof:
is -semiprimesubmodule of , let ;
+(), thus =; , and there exist
a positive integer such that , hence +
(), therefore by assumption , this implies that
=+(). For the converse, suppose that
+(); , , then . This
implies that =+().
Compare the following proposition with proposition (2.1) in
[1].
Proposition (2.11):
Let and ′ be R-modules and : ′be an
epimorphisim with .
1) If is -semiprimesubmodule of with ,
then () is -semiprimesubmodule of .
2) If is -semiprimesubmodule of , then 1() is -
semiprimesubmodule of .
Proof:
(1) () is a proper submodule of ′, since if =′,
then () for all , therefore
such that =(), this implies that
, so that = which is a contradiction and
hence () ′. Now, let and ′ with
2 +(′), since is an epimorphisim
therefore such that () = , then 2=
2=2 +(′),so that 2
+(), hence and () such
that 2=+(), then 2 (+)
therefore 2 +() , but is -
semiprimesubmodule of , this implies that and
hence = (), this means that () is -
semiprimesubmodule of′.
(2) 1(′)is a proper submodule of ′, since if
1(′) = , then ′=M=′, which is a
contradiction. Now let and such that
2 1(′) + (), therefore 2 1′+
1(′) thus 2 ′+′ but is -
semiprimesubmodule of ′, then () and hence
1(′), this implies that 1(′) is -
semiprimesubmodule of .
References
[1] Athab, E.A.1996. Prime and semiprimesubmodules,
M.SC. Thesis, Department of Mathematics, College of
Science, University of Baghdad, Baghdad, Iraq.
[2] Dauns, J.1980. Prime modules and one sided in Ring
theory and Algebra Proceedings of the Third Oklahoma
Conference Dekker, New York, 301-344.
[3] F. Kasch, Modules and Rings, Academic Press, London,
1982.
[4] D.W.Sharpe and P.Vamous, Injective modules, Lecture
in pure Math. University of Sheffield, At the univ. Press,
1972.
[5] C.P.Lu, 1984. Prime submodules of modules, Comment
Math. Univ. St. Paul,33,61-69.
Paper ID: ART20175463
DOI: 10.21275/ART20175463
1053
