A relationship between 2-primal modules and modules that satisfy the radical formula

The coincidence of the set of all nilpotent elements of a ring with its prime radical has a module analogue which occurs when the zero submodule satisfies the radical formula. A ring $R$ is 2-primal if the set of all nilpotent elements of $R$ coincides with its prime radical. This fact motivates our study in this paper, namely, to compare 2-primal submodules and submodules that satisfy the radical formula. A demonstration of the importance of 2-primal modules in bridging the gap between modules over commutative rings and modules over noncommutative rings is done and new examples of rings and modules that satisfy the radical formula are also given.


Introduction
Unless stated otherwise, all rings are unital, associative and not necessarily commutative. The modules are left unital. The set of all positive integers is denoted by N. First, we define key terms and fix notation which we later use in the sequel.
A proper ideal I of a ring R is prime (resp. completely prime) if for all ideals A, B of R (resp. a, b ∈ R) AB ⊆ I (resp. ab ∈ I), implies A ⊆ I (resp. a ∈ I) or B ⊆ I (resp. b ∈ I). Any completely prime ideal is prime but not conversely; if R is commutative, there is no distinction between the two notions. We recall a generalization of the above two ring theoretic "primes" to modules.

Submodules that satisfy the radical formula
For commutative rings, the set of all nilpotent elements of a ring R coincides with the prime radical β(R) of R which is the intersection of all prime ideals of R. In general, if I is an ideal of a ring R and √ I := {a ∈ R : a n ∈ I for some n ∈ N}, then for any ideal I of a commutative ring R we have where β(I) is the intersection of all prime ideals of R containing I. In [16], It is easy to show that if R is a commutative ring and M = R R, We say that a submodule N of an R-module M satisfies the radical formula if A module satisfies the radical formula if every submodule of M satisfies the radical formula. If every R-module satisfies the radical formula, then R is also said to satisfy the radical formula. In literature, there has been an intensive study of modules that satisfy the radical formula, see [1,2,9,12,13,17,18] among others. Unlike commutative rings for which √ I = β(I) for any ideal I, not all modules over commutative rings satisfy the radical formula.

2-primal submodules
A not necessarily commutative ring R for which √ 0 = β(R) is called a 2-primal ring. This condition forces √ 0 to be an ideal of R. It follows from [5, Proposition 2.1] that a ring R is 2-primal if and only if β co (R) = β(R), where β co (R) denotes the completely prime radical of R. We remind the reader that β co (R) is the intersection of all completely prime ideals of R and it is called also the generalized nil radical. Similarly, if I is any ideal of R, then the symbol β co (I) stands for the intersection of all completely prime ideals of R containing I. That intersection is called the completely prime radical of I. The 2-primal rings were studied by many authors (see, for example, [5,10,14,15]). An ideal I of a ring R is called 2-primal if β co (R/I) = β(R/I). ( In [8], a generalization of 2-primal rings was done to modules. A submodule N of an R-module M is 2-primal if A module is 2-primal if its zero submodule is 2-primal, i.e., if β co (M) = β(M). Any module over a commutative ring is 2-primal and a projective module over a 2-primal ring is 2-primal [8, Theorem 2.1]. As 2-primal rings bridge the gap between commutative rings and noncommutative rings, 2-primal modules also bridge the gap between modules over commutative rings and modules over noncommutative rings. 2. When does a 2-primal submodule satisfy the radical formula?

Questions to investigate
3. When does a submodule that satisfies the radical formula become 2-primal?
4. Whenever an ideal I of a ring R is 2-primal, the set √ I is an ideal of R; when does the set E M (N) become a submodule of M for a given submodule N of M?
5. Can we get modules over noncommutative rings which satisfy the radical formula?
6. Can we get noncommutative rings which satisfy the radical formula?
is not necessarily a submodule of M. Take for instance modules over a commutative ring, where each submodule is 2-primal.
In Corollary 2.7, we give a necessary and sufficient condition for a module to be 2-primal if and only if E M (0) = β(M). In Propositions 2.2 and 2.4 which have Lemmas 2.3 and 2.2 respectively as special cases, we give situations for which 2primal submodules satisfy the radical formula. Using these lemmas we are able to obtain modules and rings that satisfy the radical formula (see Theorems 2.

Main Results
Proof. Let m ∈ E M (N). Then m = rn for some r ∈ R and n ∈ M. Moreover, there exists k ∈ N such that r k n ∈ N. So, r k n ∈ β co (N). Since β co (N) is a completely semiprime submodule of M, we have m = rn ∈ β co (N). Thus E M (N) ⊆ β co (N) and finally E M (N) ⊆ β co (N).
, then x = rm and r k m ∈ N for some r ∈ R, m ∈ M and k ∈ N. As N is completely semiprime we get x = rm ∈ N.
In   (ii) a completely prime submodule of M satisfies the radical formula.

Proof.
If R is commutative, then prime submodules are completely prime. If a submodule N of M is completely prime, then it is 2-primal and prime. Hence β(N) = N and the assertion follows directly from Proposition 2.2. 3 still hold if we replace "N 2-primal" (resp. "M 2-primal") by "R is commutative". This highlights (together with the results obtained in [8]) the importance of 2-primal submodules in bridging the gap between modules over commutative rings and modules over noncommutative rings.
According to Lee and Zhou in [11], an R-module M is reduced if for all a ∈ R and every m ∈ M, am = 0 implies Rm ∩ aM = 0. An R-module is reduced in this sense if and only if for all a ∈ R and every m ∈ M, a 2 m = 0 implies aRm = 0 if and only if for all a ∈ R and every m ∈ M, am = 0 implies aRm = 0 and a 2 m = 0 implies am = 0, see [19, p.25-26]. This implies that any reduced module in the sense of Lee and Zhou is completely semiprime. A module M is symmetric if abm = 0 implies bam = 0 for a, b ∈ R and m ∈ M. An R-module M is IFP (i.e., it has the insertion-of-factor-property) if whenever am = 0 for a ∈ R and m ∈ M, we have aRm = 0. An R-module M is semi-symmetric if for all a ∈ R and every m ∈ M, a 2 m = 0 implies (a) 2 m = 0 where (a) is the ideal of R generated by a ∈ R. A submodule N of an R-module M is Lee-Zhou completely semiprime (resp. symmetric, IFP, semi-symmetric) if in the definition of reduced (resp. symmetric, IFP, semi-symmetric) we have N in the place of "0" and "∈" or "⊆" (whatever is appropriate) in the place of "=". For a detailed account of the origin of symmetric modules, IFP modules and semi-symmetric modules together with their examples, see [8].
The following chart of implications is used in the proof of Lemmas 2. Proof. From the chart of implications above it follows that any of the following implies that M is 2-primal: R is commutative, M is reduced, M is IFP, M is symmetric and M is semi-symmetric. Secondly, every free module is projective. The rest follows from Proposition 2.4 and Example 2.1.
Lemma 2.2 recovers [9, Corollary 8] which says that a zero submodule of a projective module over a commutative ring satisfies the radical formula. Proof. This follows from Proposition 2.2 and the fact that Lee-Zhou completely semiprime, IFP, symmetric or semi-symmetric submodules are 2-primal.
The following lemma was proved by McCasland and Moore in [16]. Note that, although they were working with modules over commutative rings, the proof they used still works even when the modules are not defined over a commutative ring.   Proof. If R is semisimple, then the R-module M is projective. The rest follows from Theorem 2.1.

Corollary 2.6
If R is a semisimple and commutative ring, then the R-module M satisfies the radical formula.
Proof. If R is semisimple and commutative, then M is 2-primal and projective and it is sufficient to apply Theorem 2.1.
A ring R is absolutely radical if for all R-modules M, we have β(N) = N for each submodule N of M.
Theorem 2.2 If R is an absolutely radical ring such that each submodule N of the R-module M is one of the following: Lee-Zhou completely semiprime, IFP, symmetric or semi-symmetric, then R satisfies the radical formula.
Proof. Notice that R is an absolutely radical ring if and only if β(N) = N for each submodule N of M. The rest follows from Lemma 2.3.
Proof. It follows from Proposition 2.5.
The following example shows that containment (5) in Corollary 2.7 does not hold in general.
All submodules of a module defined over a commutative ring are 2-primal but they need not satisfy the radical formula. We do not know of an example of a submodule which satisfies the radical formula but not 2-primal, although we suspect these examples exist. The motivation of our suspicion is that, for any module M, β(M) ⊆ β co (M) and E M (0) ⊆ β co (M) and these inclusions are in general strict. Hence, it is probably possible that β(M) = E M (0) ⊆ β co (M), in which case the zero submodule of M satisfies the radical formula but not 2-primal. An affirmative answer to any one of the following questions gives us the desired example(s).