International Journal of Algebra, Vol. 3, 2009, no. 3, 125 - 135 Cyclic Pure Submodules V. A. Hiremath1 Department of Mathematics, Karnatak University Dharwad-580003, India va hiremath@rediffmail.com Seema S. Gramopadhye2 Department of Mathematics, Karnatak University Dharwad-580003, India e-mail:[email protected] Abstract P.M.Cohn [3] has introduced the notion of purity for R-modules. With respect to purity, flat, absolutely pure and regular modules are studied. In this paper we introduce and study the corresponding no- tions of c-flat, absolutely c-pure and c-regular modules for cyclic pu- rity. We prove that absolutely c-pure R-modules are precisely injective modules. Also we study the relationship between c-flat and torsion-free modules over commutative integral domains and non-commutative non- integral domains. Also, we study the conditions under which c-regular R-modules are semi-simple. Mathematics Subject Classifications: 16D40, 16D50 Keywords: Pure submodules, flat module and regular ring Introduction In this paper, by a ring R we mean an associative ring with unity and by an R-module we mean a unitary right R-module. Z(M) denotes the singular submodule of the R-module M. 1Corresponding author 2The author is supported by University Research Scholarship. 126 V. A. Hiremath and S. S. Gramopadhye A ring R is said to be principal projective, if every principal right ideal is projective. We denote this ring by p.p. The notion of purity has an important role in module theory and in model theory. In model theory, the notion of pure exact sequence is more useful than split exact sequences. There are several variants of this notion. R.Wisbauer[13], generalized the notion of purity for a class ℘ of R-modules. He defines a short exact sequence 0 −→ A −→ B −→ C −→ 0ofR-modules to be ℘-pure, if every member of ℘ is projective with respect to this sequence. Cohn’s purity is precisely ℘-purity for the class ℘ of all finitely presented R- modules. With respect to this purity, absolute purity, flatness and regularity are studied. Simmons[10], considered the cyclic purity in the case of commutative integral domains. This is the ℘-purity for the class ℘ of all cyclic R-modules. Here we study the cyclic purity for general rings and also study absolute c- pure, c-flat and c-regularity. Consider a short exact sequence, :0−→ A −→ B −→ C −→ 0ofR- modules. An R-module M is said to be -injective (resp. -projective) if M is injective (resp. projective) with respect to the short exact sequence . 1. Cyclic Pure Exact sequences Definition 1.1: (i)[10] An exact sequence :0−→ A −→ B −→ C −→ 0 of R-modules is said to be cyclic pure (c-pure in short) if every cyclic R- module -projective. ii) A submodule A of an R-module B is said to be cyclic pure (c-pure in short if the canonical short exact sequence 0 −→ A −→ B −→ B/A −→ 0is c-pure. Clearly every split exact sequence is the trivial example of c-pure exact sequence. Simmons [10] considered for modules over commutative integral do- mains. He has stated in Proposition 1 (without proof) some equivalent con- ditions for cyclic purity. In the following proposition we prove that these conditions are true for general rings also. Proposition 1.2: For a submodule A of an R-module B, the following con- ditions are equivalent. i) A is c-pure submodule of B. ii) For every b ∈ B/A there exists b1 ∈ B such that b1 = b and ann(b1)= ann(b). iii) For any b ∈ B and any right ideal I of R with bI ⊆ A, there exist a ∈ A, such that (a − b)I =0. C-pure submodules 127 iv) Any system of equations xrj = aj,j ∈ J where J is any index set, with rj ∈ R is solvable in A whenever it is solvable in B. Proof: Throught the proof η denotes the canonical epimorphism from B onto B/A. i) ⇒ ii) Let b ∈ B/A and I = ann(b) then R/I is a cyclic R-module. Define, a map f : R/I −→ B/A by f(r)=br for each r ∈ R/I. Clearly it is a well-defined homomorphism. Then by (i), there exists a homomorphism g : R/I −→ B such that η ◦ g = f where η : B −→ B/A is the natural epimor- phism. Let b1 = g(1) ∈ B. Then, b1 = η(b1)=η(g(1)) = (η ◦ g)(1) = f(1) = b. So ann(b1) ⊆ ann(b). On the other hand, let r ∈ ann(b)=I. Then b1r = g(1)r = g(r)=g(o)=o and hence r ∈ ann(b1). So ann(b) ⊆ ann(b1). This proves (ii). ii)⇒ iii) Let b ∈ B and I be a right ideal of R such that bI ⊆ A. Then I ⊆ ann(b) where b = b + A. Now by (ii), there exists b1 ∈ B such that ann(b)=ann(b1) and b = b1 . Hence b − b1 = a ∈ A. Since I ⊆ ann(b)=ann(b1)=ann(b − a), this implies (b − a)I = o. iii)⇒iv) Let xrj = aj,j ∈ J (where J is any index set) (∗), be a system of equations with rj ∈ R and aj ∈ A have solution in B. Then there exists b ∈ B such that brj = aj for every j ∈ J. Let I be the right ideal of R generated by the { } ⊆ ∈ subset rj j∈J of R.NowbI A. Then by (iii) there exists a A such that, (b − a)I=o. This implies that (b − a)rj = o for every j ∈ J. So, brj = arj for every j ∈ J. Hence (∗) has solution in A. iv)⇒i) Let M be a cyclic R-module. We may assume that M = R/I for some right ideal I of R. Let f ∈ Hom(R/I, B/A) and let b = f(1). Consider the subset { ∈ ∈ } { } J = r R/br A of R which we write rα α∈Λ. Let brα = aα for every α ∈ Λ. Hence the system of equations xrα = aα has solution b in B.By (iv), there exists a ∈ A such that arα = aα for every α ∈ Λ. Now we de- fine a map g : R/I −→ B by, g(r)=(b − a)r for every r ∈ R/I.Ifr ∈ I, ∈ ∈ { } br = br = f(1)r = f(r)=f(o)=o and hence br A.Sor J = rα α∈Λ. ∈ − − − Let r = rα0 , for some α0 Λ. So (b a)r =(b a)rα0 = brα0 arα0 = − aα0 aα0 = o. Hence g(r)=o. Hence g is a well-defined homomorphism. Also (η ◦ g)(r)=η((b − a)r)=br = f(r), since, ar ∈ A = ker(η). Hence η ◦ g = f. Hence the result. Corollary 1.3: Let A be a c-pure submodule of an R-module B. i) If B is torsion-free, so is B/A. ii) If B is non-singular, so is B/A. Proof: i) Let br = o for some b ∈ B/A and some r ∈ R. By Proposition 1.2 ii) there exists b1 ∈ B such that ann(b)=ann(b1) and b = b1. Since, r ∈ ann(b), r ∈ ann(b1). By hypothesis, B is torsion-free, implies there is no 128 V. A. Hiremath and S. S. Gramopadhye regular element in ann(b1). Hence, r cannot be regular. So, B/A is torsion-free ii) Let x ∈ Z(B/A). Then ann(x) is an essential right ideal of R. Since, A is c-pure in B, by definition, there exists, y ∈ B such that, ann(x)=ann(y) and x = y. Since, x ∈ Z(B/A), y ∈ Z(B). Since, B is non-singular, Z(B)=0 and hence y = 0. So, x = y = 0. Hence, Z(B/A) = 0, which implies B/A is non-singular. Now we give some simple properties of cyclic pure submodules. The proofs are straightforward and hence omitted. Proposition 1.4: Let A, B, C be R-modules such that A is a submodule of B and B is a submodule of C. i) If A is c-pure in B and B is c-pure in C then A is c-pure in C. ii) If A is c-pure in C then A is c-pure in B. iii) If B is c-pure in C then B/A is c-pure in C/A. iv)If A is c-pure in C and B/A is c-pure in C/A then B is c-pure in C. Now we will give one more characterization for c-pure submodules, which will be used later. Proposition 1.5: A submodule A of an R-module B is c-pure submodule of B if and only if for every submodule C of B containing A, with C/A cyclic, A is a direct summand of C. Proof:Only if: Let A be a c-pure submodule of an R-module B and let C be a submodule of B containing A such that C/A cyclic. Then by Proposition 1.4(ii) above, A is a c-pure submodule of C. Then C/A, being cyclic, is projective with respect to the canonical short exact sequence, 0 −→ A −→ C −→ C/A −→ 0 and hence A is direct summand of C. If: Consider the canonical short exact sequence :0−→ A −→ B −→η B/A −→ 0ofR-modules. Let M be a cyclic R-module and let f ∈ HomR(M,B/A). Now f(M) is a submodule of B/A and being homomorphic image of a cyclic R-module, it is also cyclic.