On completely prime submodules

The formal study of completely prime modules was initiated by N. J. Groenewald and the current author in the paper; Completely prime submodules, {\it Int. Elect. J. Algebra}, {\bf 13}, (2013), 1--14. In this paper, the study of completely prime modules is continued. Firstly, the advantage completely prime modules have over prime modules is highlited and different situations that lead to completely prime modules given. Later, emphasis is put on fully completely prime modules, (i.e., modules whose all submodules are completely prime). For a fully completely prime left $R$-module $M$, if $a, b\in R$ and $m\in M$, then $abm=bam$, $am=a^km$ for all positive integers $k$, and either $am=abm$ or $bm=abm$. In the last section, two different torsion theories induced by the completely prime radical are given.


Introduction
Completely prime modules were first formally studied in [8] as a generalization of prime modules. These modules had earlier appeared informally in most cases as examples in the works of: Andrunakievich [1], De la Rosa and Veldsman [5, p. 466, Section 5.6], Lomp and Peña [12,Proposition 3.1], and Tuganbaev [18, p. 1840] which were published in the years 1962, 1994, 2000 and 2003 respectively. In [1] and [5] these modules were called modules without zero-divisors, in [12] they were not given any special name and [18] they were called completely prime modules. In this paper just like in [8], we follow the nomenclature of Tuganbaev. A submodule P of M is a completely prime (resp. completely semiprime, prime) submodule if the factor module M/P is a completely prime (resp. completely semiprime, prime) module. A completely prime module is prime but not conversely in general. Over a commutative ring, completely prime modules are indistinguishable from prime modules.
Example 1. 1 We know that the ring R = M n (F) of all n × n matrices over a field F is prime but not completely prime, i.e., it is not a domain. Since for a unital ring R, R is prime (resp. completely prime) if and only if the module R R is prime (resp. completely prime), see [8,Proposition 2.4], we conclude that the R-module R (where R = M n (F)) is prime but not completely prime.

A road map for the paper
This paper contains five sections. In Section 1, we give an introduction, define some of the notation used and describe how the paper is organized. In Section 2, we state the advantage completely prime modules have over prime modules. They behave as though they are defined over a commutative ring, a behaviour prime modules do not have in general. The aim of Section 3 is two fold; we provide situations under which a module becomes completely prime and furnish concrete examples for completely prime modules. In Section 4, we define completely co-prime modules by drawing motivation from how prime, completely prime and co-prime modules are defined. A chart of implications is established between completely co-prime modules, co-prime modules, completely prime modules and prime modules, see Proposition 4.2. In Proposition 4.2 it is established that the notion of completely co-prime modules is the same as that for fully completely prime modules, i.e., modules whose all submodules are completely prime. Many other equivalent formulations for completely co-prime modules are given. It is shown that if M is a fully completely prime R-module, then for all a, b ∈ R and every m ∈ M, abm = bam, am = a k m for all positive integers k, and either am = abm or bm = abm. In Section 5, which is the last section, we give two torsion theories induced by the completely prime radical of a module. On the class of IFP modules (i.e., modules with the insertion-of-factor property), the faithful completely prime radical is hereditary and hence leads to a torsion theory, see Theorem 5.1. Lastly, we show in Theorem 5.2 that the completely prime radical is also hereditary on the class of semisimple R-modules and therefore it induces another torsion theory.
2 Advantage of completely prime modules over prime modules Where as prime modules form a much bigger class than that of completely prime modules, completely prime modules possess nice properties which prime modules lack in general. Completely prime modules over noncommutative rings behave like modules over commutative rings. In particular, they lead to the following properties on an R-module M: P1. for all a, b ∈ R and m ∈ M, abm = 0 implies bam = 0; P2. for all subsets S of M and m ∈ M \ S, (S : m) is a two sided ideal of R; P3. for all a ∈ R and m ∈ M, am = 0 implies arm = 0 for all r ∈ R; P4. the prime radical of M coincides with its completely prime radical, i.e., the intersection of all prime submodules of M coincides with the intersection of all its completely prime submodules.
A module which satisfies property P1, P3 and P4 is respectively called symmetric, IFP (i.e., has insertion-of-factor property) and 2-primal. Properties P2 and P3 are equivalent. To prove the claims made in this section, one only needs to prove the following implications for a module: completely prime ⇒ completely semiprime ⇒ symmetric ⇒ IFP ⇒ 2-primal, see [9, Theorems 2.2 and 2.3] and [6] for the proof. A submodule P of an R-module M is said to be symmetric (resp. IFP) if the module M/P is symmetric (resp. IFP).
A comparison with what happens for rings indicates that these results on modules are what one would expect. Every domain (completely prime ring) is reduced (i.e., completely semiprime) so it is symmetric, IFP and 2-primal, see [13]. Note that the IFP condition is called SI in [13]. The notions of IFP and symmetry first existed for rings before they were extended to modules.  According to Reyes [15, Definition 2.1], a left ideal P of a ring R is completely prime if for any a, b ∈ R such that P a ⊆ P , ab ∈ P implies that either a ∈ P or b ∈ P . In Proposition 3.2, we characterise completely prime submodules in terms of zero divisor sets of their factor modules.    Proof: Suppose am = 0 for some a ∈ R and m ∈ M. If m = 0, M is a completely prime module. Suppose m = 0. Then am = a n i=1 r i m i = n i=1 (ar i )m i = 0 for some r i ∈ R and m i ∈ M with i ∈ {1, 2, · · · , n}. M being free implies ar i = 0. m = 0 implies there exists j ∈ {1, 2, · · · , n} such that r j = 0. ar j = 0 implies a = 0 since R is a domain and r j = 0. Hence, aM = {0} and M is completely prime. These definitions motivate us to define completely co-prime modules.    For if am ∈ P for some a ∈ R, m ∈ M and P ≤ M, we get aR ⊆ (P : m) since (P : m) is a two sided ideal as R is left-duo. 1 So, aRm ⊆ P . By hypothesis, P is a prime submodule of M, hence m ∈ P or aM ⊆ P which proves that P is a completely prime submodule.

Completely co-prime modules
If R is a commutative ring, then fully prime R-modules are indistinguishable from fully completely prime modules. Fully prime modules over commutative rings were studied in [4]. Example 4.3 Fully completely prime rings were studied by Hirano in [11]. If R is a fully completely prime ring such that R has no one sided left ideals, then the module R R is a fully completely prime module.
A module is fully IFP if all its submodules are IFP submodules.

Proposition 4.3 A cyclic module over a fully completely prime ring is fully completely prime.
Proof: We use the fact that a fully completely prime ring is fully IFP. Let M = Rm 0 , N ≤ M and am ∈ N for some a ∈ R and m ∈ M. Then arm 0 ∈ N for some r ∈ R where m = rm 0 . ar ∈ (N : m 0 ). Since R is fully IFP, (N : m 0 ) is a two sided ideal. Thus, a ∈ (N : m 0 ) or r ∈ (N : m 0 ) by hypothesis so that aRm 0 ⊆ N or rm 0 ∈ N. From which we obtain aM ⊆ N or m ∈ N.  A ring is said to be a chain ring if its ideals are linearly ordered by inclusion. A chain ring is sometimes called a uniserial ring.   4. the faithful completely prime radical is idempotent, i.e., (β f co ) 2 = β f co . Proof: 1. Suppose N is an essential submodule of an R-module M such that N is a faithful completely prime module. We show that M is also faithful and completely prime. Let a ∈ R and m ∈ M such that am = 0. If m = 0, M is completely prime. Suppose m = 0. Since N is an essential submodule of M, there exists r ∈ R such that 0 = rm ∈ N. am = 0 implies arm = 0 since by hypothesis we have a class of IFP modules. N completely prime together with the fact that 0 = rm ∈ N lead to a ∈ 3. the completely prime radical is idempotent, i.e., β 2 co = β co . Proof: