Almost Pure Ideals (Submodules) and Almost Regular Rings (Modules)

Let R1be a commutative2ring with identity and M be a unitary R-module. In this6work we7present almost pure8ideal (submodule) concept as a9generalization of pure10ideal (submodule). Also, we1generalize some9properties of8almost pure ideal (submodule). The 7study is almost regular6ring (R-module).


Introduction
Let M be a1unitary2R-module and let9R be commutative8ring with identity. A submodule N of an7R-module M is called6pure if N∩ , for each5ideal of a ring4R", E, Anderson and Fuller [1]. Fieldhous (1969) in [2] defined N to be pure in M if for each And defined a ring R is regular if every ideal I is pure. Although Fieldhous generalize regular rings to regular modules in [2], Ware (1971) in [3] and Zelmanowliz (1972) in [4], also study regular modules. Nuhad S. Al-Mothafar and Ghaleb A. Humod in [5] study 2-pure submodules and 2-regular modules they define N to be 2-pure in M if N∩I 2 M=I 2 N, for each ideal in R.
In this paper we5introduce an almost pure ideals (submodules) concept as a generalization of pure ideals (submodules). An ideal of6a ring R is said to be almost pure if ∩ = where is the Jacobson radical of R. A submodule N of an R-module M is called almost pure if N∩ M= N, we study some properties of almost pure submodules. Also, we get some results of almost pure submodules like a (Prop. 3.4) let M be an R-module and N be an almost pure submodule of M. If S is a submodule of M containing N, then N is an almost pure submodule of S.
And we present almost regular rings (modules) concepts as generalization of regular rings (modules) we generalize some properties of regular ideals (modules) to almost regular rings (modules).
This work includes three sections. In section two, we introduce almost pure ideals and almost regular rings concepts as a generalization4of pure ideals and regular rings, we provided in (Prop. 2.9). If R is A-regular integral domain such that , then R is semisimple ring.

ISSN: 0067-2904
In section three, we introduce almost pure submodules and almost regular modules concepts as a generalization of pure submodules and regular modules, we present (Prop. 3.16) let M be an almost regular R-module9contain a non torsion element, then R is almost regular ring. Finally in section three, we0see in proposition (3.15) that if M is almost regular R-module, then ∀ 0≠x M such that ann(x)  , then R/ann(x) is almost regular ring.

Almost Pure Ideals and Almost Regular Rings
In this section, we0introduce a generalization for pure ideals concept namely almost pure ideal. First we "recall that an ideal I of a ring R is called pure if for each ideal of R. It is well know that a ring R is (Von Neumann) regular if and only if every ideal of R is pure , equivalently for every there exist such that [2]. We start by the following definition .

Definition (2.1):
An ideal of a ring R is called almost pure ( for short A-pure ) ideal of R if where is the Jacobson radical of .

Remarks and examples (2.2):
1. It is clear0that every pure ideal of a ring is A-pure, but the3converse is not true in general, for example: In a ring Z, the ideal I= < 3 > is A-pure, since < 3 > ∩ {0} = < 3 > .{ 0}, but < 3 > is not pure in the ring Z. 2. Every8ideal generated by idempotent is A-pure. "Since every ideal that generated by idempotent is pure [6], hence is A-pure by remark (1).

If
are A-pure ideals of a ring R, but need not be A-pure ideal of R. To show that if and are A-pure ideals of R but In the ring Z every ideal is A-pure. 5. It is clear that and are always A-pure ideals of any ring R. "Recall0that a ring R is called regular if every ideal in R is pure", [2]. Now, we have the following:

Definition (2.3):
A ring R is called almost regular (for short A-regular) if every ideal in R is A-pure. (i.e.) ∩ = (R) ∀  R.

Remarks and examples (2.4):
1-Every0regular ring is A-regular. But the converse is not true in general the ring Z is A-regular. Since (Z) =0, but not regular.

Theorem (2.5):
A7ring R is A-regular if and only if is pure ideal in R.

Proof
Let be an ideal of R. Since R is A-regular, hence is A-pure in R. (i.e) ∩ = . Thus ) is pure in R. The converse is clear.

Proposition (2.6)
Let :R Rʹ be an R-epimorphism such that If R is A-regular ring, then Rʹ is Aregular ring . Proof : Let Iʹ be an ideal in a ring Rʹ, we have to show Iʹ ∩ J(Rʹ )= Iʹ J(Rʹ). Now, Iʹ = (I), where I is an ideal in a ring R, since R is A-regular ring then Since is an epimorphism whith then ( ) since is epimorphism and But since is epimorphism and , [7 ]. Thus

Proposition (2.7)
Let R be a ring and I be an9ideal of R contained in J(R). If R/I is A-regular ring, then R is Aregular.

Proof
To show R is A-regular ring, we have to show K∩ )=K for all ideal K of R, since is A-regular ring and is an ideal of , then ) = )) since ⁄ is an epimorphism and then ( ) ( ) [7 ]. hence ∩ = , then = s  . .

Proposition (2.8):
A finite direct sum of A-regular rings R i such that i  J(R) ∀ is A-regular.

Proof
Let {R i } , i=1,2,…,n be a finite number of A-regular ring. Let R=R 1 R 2 …R n be their direct sum, to show R is A-regular. Let n=2 , then R=R 1 R 2 , It is clear then R/R 1 R 2 , since R 2 is Aregular by proposition(2.6) R/R 1 is A-regular, hence by proposition(2.7) R is A-regular. Using8mathematical induction. Assume that the statement is true for n=k to show that it is true for k+1, (i.e) R=R 1 … R k is A-regular, and to show that R= R 1 … R k R k+1 is A-regular since ⁄ and is A-regular , then R/ R 1 … R / R k is A-regular. proposition (2.7) implies R=R 1 R 2 … R k R k+1 is A-regular. " Recall8that if R is integral domain then R is pure simple, where R is pure simple if it has no pure ideal exept {0} and R , [8]. We know that if R is regular integral domain , then R is a field. But this is not true for A-regular, since Z is A-regular integral domain but not field.

Proposition (2.9)
If R is A-regular integral domain such that , then R is semisimple ring. proof: Since8R is A-regular, then by (2.8) is pure ideal in R. But R is integral domain, hence R has no pure ideal exept {0} and R. Since , hence . Thus R is semisimple Proposition (2.10) If is a prime ideal in R and R is A-regular ring such that , then R/ is semisimple ring.

Proof
Since9 is prime ideal in R/ integral domain, and by (2.6) R/ is A-regular. Since , then by proposition (2.9 ) R/ is semisiple.

Almost Pure Submodules and Almost Regular Modules
" Recall9that a sub module N of an R-module M is called pure if , for each ideal of a ring R, [1]. We start this section by the following definition.

Definition (3.1)
A submodule N of an R-module M is called almost pure, (for short A-pure), if N ∩ J (R) M = J (R) N, where J (R) is the Jacobson radical of a ring R.

Remarks and Examples (3.2)
1. It ' s clear9that every pure submodule is A-pure, but the converse is not true in general. 3. It is clear9that every submodule in Z as Z-module is A-pure. 4. It ' s clear9that every direct summand of any R-module M is A-pure submodule of M since every direct summand is pure summodule, hence is A-pure by Remark(1), but the converse is not true, for example the submodule { 0 , 3 , 6}of the module Z 9 as Z-module is A-pure since { 0 , 3 , 6}∩ J(Z)Z 9 =J(Z) { 0 , 3 , 6} but not direct summand.

Proposition (3.3)
Let M be an R-module and N be an8 A-pure submodule of M. If S is A-pure in N, then S is an Apure submodule of M.

Proof
Since

Proposition (3.4)
Let M be an R-module and N be an A-pure submodule9of M. If S is a submodule of M containing N, then N is an A-pure submodule9of S.

Proof
Since " Recall9that an R-module M is called an F-regular module if every submodule of M is pure", [9]. Now, we have the following definition.

Definition (3.6)
An9R-module M is called A-regular module if every submodule of M is A-pure.

Remarks and Examples (3.7):
1. It is clear9that every F-regular is A-regular R-module but the convers is not true in general. For example Z 4 as Z-module is A-regular since every submodule of Z 4 is A-pure but Z 4 is not F-regular, see remark and examples (3.2 ) (1). 2. The Z-module Z is A-regular since every submodules9of Z is A-pure. But Z as Z-module is not Fregular.
3. The Z 9 as Z-module is A-regular since every submodule of Z 9 is A-pure. But Z 9 is not F-regular since { 0, 3, 9} is not pure, see remark and examples (3.2) (1). 4. The Z 12 as Z-module is A-regular since every submodule of Z 12 is A-pure, but Z 12 is not F-regular since the submodule < 2> is not pure. 5. Q as Z-module is A-regular since Q∩ (Z)Q= (Z)Q, but Q as Z-module is not F-regular.

Theorem (3.8):
Let M be an9R-module. The following statements are equivalent: 1. M is A-regular module. The following proposition shows that the factor module of A-regular module is A-regular.

Proposition (3.9):
Let M be an R-module, then M is A-regular if and only if M/N is A-regular for every submodule9N of M.

Proof
Let N be a submodule9of an R-module M and K be any submodule of M containg N. Since M is Aregular, then K is A-pure in M, hence K/N is A-pure in M/N proposition (3.5). Thus M/N is A-regular. The converse is easily by taking N=0.

Corollary (3.10)
Let M and Mʹ be R-modules and ƒ: M Mʹ is an R-epimorphisim. If M is A-regular R-module, then Mʹ is A-regular.

Proof
Since9ƒ:M Mʹ be an Repimorphisim and M is A-regular R-module, then M/kerf is A-regular Rmodule by proposition (3.9). But M/kerf is isomorphic to Mʹ by the first isomorphism theorem. Thus Mʹ is A-regular.

Corollary (3.11)]
Every9submodule of A-regular module is A-regular module.

Proof
Let N be a submodule of an A-regular M. To show that N is A-regular R-module, let K be any submodule of N. Since M is A-regular. Hence we have: Recall9that an R-module M is called F-regular if and only if ∀ x M and ∀ r ∃ t R such that rx=rtrx", [6].

Preposition (3.12):
If M is A-regular R-module, then for every nonzero element x in M and for each r (R), there exit t R such that rx = rtrx.

Proof
Let and Since and implies A-regular.
. Hence , which implies that where Preposition (3.13): let M be an R-module is each nonzero element x in M and for every r (R), for some then M is A-regular. Proof let N be a submodule of a module M, to show . Let then and , hence where r (R) and m , for some Thus which implies that M is A-regular. corollary (3.14) let R be an A-regular ring, then for each for some

Proposition (3.15)
If M is A-regular R-module, then ∀ 0≠x M such that ann(x)  ,then R/ann(x) is A-regular ring.

Proposition (3.16)
Let M be an A-regular R-module contains a9non torsion element, then R is A-regular ring.