Semi – Bounded Modules

Let R be a commutative ring with identity, and let M be a unity R-module. M is called a bounded R-module provided that there exists an element xM such that annR(M) = annR(x). As a generalization of this concept, a concept of semi-bounded module has been introduced as follows: M is called a semi-bounded if there exists an element xM such that R R ann M ann (x)  . In this paper, some properties and characterizations of semi-bounded modules are given. Also, various basic results about semi-bounded modules are considered. Moreover, some relations between semi-bounded modules and other types of modules are considered.


Introduction:
Let R be a commutative ring with unity, and let M be an R-module. An R-module M is called a semi-bounded module if there exists an element xM such that RR ann M ann (x)  , where ann R M={r:rR and r m = 0 , mM}. Our concern in this paper is to study semi-bounded modules and to look for any relation between semibounded modules and certain types of well-known modules especially with bounded modules. This paper consists of two sections. In the first section, the definition of a semi-bounded module is recallded and we illustrate it by some examples, we also give some of the basic properties of semi-bounded modules. We end the section by studying the localization of semibounded modules, see (1.13).
In section two, we study the relation between semi-bounded modules and bounded modules. It is clear that every bounded module is semi-bounded module, but the converse is not true in general. We give in (2.1), a conditions under which the two concepts are equivalent. Next we investigate the relationships between semi-bounded, prime, quasi-Dedekined, cyclic and multiplication modules see (2.3), (2.8).1.
Semi-Bounded Modules Following [1] an R-module M is said to be a bounded module if there exists an element xM such that ann R M = ann R (x), where ann R M = {r  R; r m=0,mM}. In this section the concept of semi-bounded module is introduced as a generalization of a bounded module and we give some properties and characterizations for this concept. We end this section by studying the behaviour of semibounded modules under localization.
We claim that . Let r R ann ((x, y)) , then r n (x,y)=(0,0) for some nZ + , and so (r n x,r n y)=(0,0). It follows that r n x=0 and r n y=0, that is r n ann R (x) and r n ann R (y) and so  Recall that an R-module M is called F-regular if every submodule of M is pure, [4]. Proof: It is enough to show N is pure in M. Since N is a divisible submodule of M, then rN=N for every 0rR and so rMN=rMrN=rN. Thus N is pure. Therefore M is a semi-bounded Rmodule by proposition (1.8).Recall that a subset S of a ring R is called multiplicatively closed if 1S and abS for every a, bS, we know that a proper ideal P in R is prime if and only if R\P is multiplicatively closed, [5 ,p.42]. Now, let M be an R-module and S be a multiplicatively closed subset of R and let R S be the set of all fractional r/s where rR  and s S and M S be the set of all fractional x/s where x  and sS. For x 1, x 2   and s 1, s 2  S, x 1 /s 1 = x 2 /s 2 if and only if there exists tS such that t(s 1 x 2s 2 x 1 )= 0 . So, we can make M S in to R s -module by setting x/s+y/t=(tx+sy)/st and r/tx/s=rx/ts for every x,yM and s, tS, rR  . and M S is the module of fractions. If S=R\P where P is a prime ideal we write M p instead of Ms and R p instead of R s . R p is often called the localization of R at P, and M p is the localization of M at P. In order, we investigate the behaviour of a semi-bounded module under localization. But first we state and prove the following lemma. Lemma 1.12: Let M be an R-module, let S be a multiplicatively closed subset of R. If ann (x) =(ann R (x)) S for some nZ + . Hence, there exists r 1 ann R (x) and tS such that r n /s n =r 1 /t and so there exists t 1 S such that t 1 tr n =t 1 r 1 s n ann R (x), which implies that t 1 tr n ann R (x) and so The following corollary follows immediately from proposition (1.13). If P is a prime ideal of R and M is a finitely generated semi-bounded Rmodule, then M P is a semi-bounded R P -module.

Some Relations Between Semi-Bounded Modules and Other Modules
In this section, we study the relationships between semi-bounded modules and other modules such as bounded modules, prime, quasi-Dedekined, cyclic and multiplication modules. As we have mentioned in (1.2(1)), that bounded module is a semi-bounded module and the converse need not be true in general. However the following result shows that the converse is true. But first the following definition is needed.Recall that a submodule N of an R-module M is said to be semi-prime if for every rR, xM, kZ + , such that r k xN, then rxN, see [7]. Hence, r R ann M , which implies that r n ann R (M) for some nZ + . Thus, r n m=0 for each mM. But (0) is a semi-prime submodule of M, then rm=0 and hence rann R (M), so that ann R (x)ann R (M). Therefore, ann R (x)=ann R (M), that is M is bounded R-module. Next, we study the relationship between semi-bounded modules and prime modules. And we give a condition under which the two concepts are equivalent. Recall that an R-module M is said to be prime module if ann R M=ann R N for every non-zero submodule N of M, [8]. It is clear that every prime R-module is bounded and hence it is semi-bounded, but the converse need not be true in general, for example: Let M=Z 8 as a Z-module is bounded and so semi-bounded, but not prime module since ann Z Z 8 =8Z but ann Z (2) =4Z. In order we can give the following result. But first we need the following definition. Recall that a submodule N of an R-module M is called a bounded if there exists xN such that ann R N=ann R (x), see [2]. Proposition 2.2: Let M be an R-module and let 0xM such that: If M is a uniform semi-bounded R-module such that (0) is a semi-prime submodule of M and every non-zero submodule of M is bounded, then M is a quasi-Dedekind. Proof: By proposition (2.3), M is a prime R-module. But M is uniform, so by [9,theorem 11,ch.2] we obtain the result. As we mentioned in (1.2,(6)), that cyclic module is a semi-bounded and the converse need not be true in general. However the following result shows that the converse is true. But first the following definition is needed. Recall that an R-module M is said to be fully stable if ann M (ann R (x))=(x) for each xM. [10,corollary 3.5]. In the following proposition, we give a condition under which the converse of (1.2,(6)) is true. Proposition 2.6: If M is a fully stable semi-bounded Rmodule and (0) is a semi-prime submodule, then M is cyclic R-module. Proof: Since M is a semi-bounded Rmodule and (0) is a semi-prime submodule, then M is a bounded by proposition (2.1). But M is a fully stable, so by [2,proposition 1.1.4,ch.1] we obtain the result. Now, the relationship between semi-bounded modules and multiplication modules has been studied. And we give a condition under which the two concepts are equivalent. Recall that an R-module M is said to be multiplication module if for every submodule N of M, there exists an ideal I in R such that N=I M, [11]. Note that it is not necessary that every semi-bounded is multiplication for example: Q as a Z-module is semibounded, but not multiplication, since Z is a submodule of Q, but  I an ideal of Z such that IQ=Z.
In the following corollary, we give a sufficient condition for semi-bounded module is multiplication. Corollary 2.7: If M is a fully stable semi-bounded Rmodule and (0) is a semi-prime submodule, then M is a multiplication R-module. Proof: By proposition (2.6), we obtain that M is a cyclic R-module. Then it is clear that M is a multiplication Rmodule. In the following proposition, we give some condition under which the converse of corollary (2.7) is true. But first we need the following definition. Recall that an R-module M is called a quasi-prime R-module if and only if ann R N is a prime ideal for each non-zero submodule N of M, [6]. Proposition 2.8: If M is a multiplication quasi-prime Rmodule, then M is a semi-bounded Rmodule. Proof: Since M is a multiplication quasi-prime R-module, so M is a prime module by [6,theorem 1.4.1,ch.1], hence it is a bounded. Therefore M is a semi-bounded R-module by (1.2,(1)).