Apr 4, 2015 - Dummit RM. Foote, Abstract Algebra 3rd edition John Wiley and. Sons, Inc 2004. [5] JB Fraleigh V J Katz, Basic Abstract Algebra 7thedition, ...
International Journal of Algebra, Vol. 9, 2015, no. 3, 147 - 153 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2015.512
T -Noncosingular Abelian Groups Surajo Sulaiman Northest University, Kano Department of Mathematics, Faculty of Sciences PMB 3220 Gidan Ado Bayero Kofar Nassarawa Kano-Nigeria c 2015 Surajo Sulaiman. This is an open access article distributed under the Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract In this paper, we study the notion of T -noncosingular abelian groups, which means an abelian group whose nonzero endomorphisms are not small. We show that injective (divisible) and projective (free) groups are T -noncosingular. We proved that T -noncosingular torsion groups are exactly the direct sum of a semisimple group C and a divisible group D which does not have simple subgroups isomorphic to a subgroup of C. Many other interesting results are presented here, we also give some condition for torsion-free groups to be T noncosingular (Though torsion-free groups are K-noncosingular).
Keywords: Abelian group, Torsion group, Torsion-free group, T-noncosingular, small homomorphims, small subgroup, simple group and semi-simple group
1. INTRODUCTION In 2009 Derya Keskin Tutuncu and Rashid Tribak introduced and studied the concept of T -noncosingular Modules[13] and their work was due to the concept (which is a dual) of K -nonsigular modules and application presented by S.T Rizvi and C.S. Roman[10] in 2007. The actual concept of K-noncosingular was introduced by Rizvi and Roman in the paper Bear and Bear modules in 2004 and this paper was from the Doctoral Dissertation of Roman C. S. (2004) in Ohio state university. Also in 2013 Rashid Tribak presented some result on
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T -noncosingular Modules [14]. In 2010 Ozan Gunyuz also in his MSc thesis presented and studied some further notion which they defined as Strongly T -noncosingular Modules. In the view of the above we present the notion of T -noncosingular Abelian groups and since an abelian group is a Z -module, we shall use most of the definitions and properties of modules satisfying Z -modules for the Abelian groups. for under notation and other definition the reader can see [2], [3],[4],[5], [6],[7] and[9]. 2. T -NONCOSINGULAR ABELIAN GROUPS Definition 0.1. A subgroup H of an abelian group G is called small or (superfluous) in G if for all subgroup K of G,then equality H+K=G implies K=G. ( if H is a small subgroup of a group an abelian G then we write H G) (see [8]) Definition 0.2. Let G and H be two Abelian groups. We say that G is T noncosingular relative to H, if for every 0 6= f Hom(G, H), the Imf is not small in H. Definition 0.3. Let G be an abelian group. We say that G is a T -noncosingular abelian group if it is T -noncosingular relative to itself, that is for every 0 6= f E = End(G), the Imf is not small in G. In other words G is T -noncosingular if and only if for every nonzero endomorphism f of E,Imf G implies that f = 0. From the definition 0.2 and 0.3 above we can clearly see that every noncosingular is also T -noncosingular Abelian group; however we can see that Zp is T -noncosingular but not noncosingular which means the converse need not be true. Really for a nonzero endomorphism f : Zp −→ Zp2 , defined byf (k) = pk, we have Imf =< p > Zp2 Proposition 0.4. Every simple group S is T -noncosingular. Proof; For every 0 6= f E = End(G), Imf ≤ S and S is simple this means that Imf = S. This means that Imf is not small in S Following [9, 21.2 (Baer)] we already know that divisible groups are injective groups and the image of a divisible group is a direct summand, we can now state the following. Proposition 0.5. Every divisible group D is T -noncosingular. Proof; for every 0 6= f : D −→ D, this means that Imf is also divisible and hence a direct summand of D this means D = Imf ⊕ K , for some subgroup K, this means that Imf is not small in D T Remember that for an abelian group G the radical of a group G is RadG = pG, where p runs over all prime integers.
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Proposition 0.6. If RadG = 0 then G is T -noncosingular. Proof: suppose that Imf G, for an endomorphism f : G −→ G. Then Imf ≤ RadG = 0, therefore. Imf = 0 that isf = 0 by definition 0.3 and so G is T -noncosingular. T Note that Rad Z = pZ = 0, we can state the proposition below; Proposition 0.7. Z is T - noncosingular abelian group. Proof: follows from proposition 0.6 n Q : m is a square free,m = p1 .p2 .....pk }, then A Corollary 0.8. Let A = { m is T - noncosingular. Proof: LetTf : A −→ A be an endomorphism with Imf A . Then Imf ≤ RadA = pA = Z where the intersection is taken over all prime numbers p. On the other hand 1Z = RadA, therefore f (1)f (RadA) ⊆ Radf (A) = RadImf ≤ RadZ = 0. So we see that 1Kerf, henceZ ⊆ Kerf. Then for n n n A, we have mf ( m ) = f (m m ) = f (n) = 0. But A is torsion free, every m n hencef ( m ) = 0.Sof = 0
We have been mentioning different abelian groups which are T-noncosingular, let us at this point state some useful examples. Example 0.9. Zpn is not T -noncosingular abelian group, for any integer n > 1, and prime p Proposition 0.10. Let G be a T -noncosingular abelian group and H be a direct summand of G, then H is also T - noncosingular. Proof:- Let G = H ⊕ K and define f : H −→ H with Imf H then consider the homomorphism g = f ⊕ 0 : G −→ G defined by g(h, k) = (f (h), 0). Then Imf = Imf ⊕0 H ⊕K. Since G is T -noncosingular, g = 0 therefore f = 0. see [13] Above result shows that direct summand of T -noncosingular is also T noncosingular, the natural question here is that, what about direct sum of T -noncosingular? This question and other results will be treated in the next section. 3. DIRECT SUMS OF T-NONCOSINGULAR ABELIAN GROUPS The following example will answer our question of the previous section and look at the condition that may generalised the answer to the problem. Example 0.11. We have seen above that both Zp and Zp ( Zp is divisible and divisible groups are T -noncosingular) are T -noncosingular. We will now show that their direct sum G = Zp ⊕ Zp is not T -noncosingular.
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Really, define f : G −→ G by f (k, cn ) = (0, kc1 ). clearly f is a homomorphism and Imf = 0⊕ < c1 >. since < c1 > Zp , then Imf G and of course f 6= 0. So G is not T -noncosingular. The following proposition gives the condition for which direct sum of T noncosingular abelian group to be T -noncosingular. Proposition 0.12. Let (Hi )(iI) be a family of subgroups of G, and G = L (Hi )(iI) , then G is T -noncosingular if and only if Hi is T - noncosingular related to Hj for all i, jI. L Proposition 0.13. Let G = Hi , G is T -noncosingular if RadHi = 0 for each i. Proof: follows from proposition 0.6 Note that from the propositionb 0.12 and 0.13 above we can draw an important result as follow Corollary 0.14. Every L semisimple group C is T -noncosingular. Proof; C = ΣZp = Zp where p is prime, if p 6= q then Hom(Zp , Zq ) = 0, so Zp is T -noncosingular related to Zq by proposition 0.12 C is T -noncosingular. Corollary 0.15. Every free group F is T -noncosingular Proof:we know from [11], F = ΣxX < x > with each < x >∼ = Z where X is a basis of F, therefore we can write F = ΣZ. and each Z is T -noncosingular related to itself from proposition 0.13, hence F is T -noncosingular. Theorem 0.16. A torsion group G is T -noncosingular if and only if G = L D C, where D is divisible and C is semi-simple and if C has a direct summand isomorphic to Zp for some prime p, then D has no direct summand isomorphic to Zp (That is if Cp 6= 0 then Dp = 0). P roof : (−→) Let G be a torsion T -noncosingular group and D be its maxL imal divisible subgroup. Then G = D C for some C ≤ G. Let p be prime and B be the basic subgroup of p component Cp of C. If B has a direct summand M isomorphic to Zpn with n > 1, then since B is a pure subgroup of Cp , then M is a pure subgroup of G, hence M is a direct summand of G by [11, 10.41], therefore is T - noncosingular. But we know that Zpn with n > 1, is not T -noncosingular. So B is the direct sum of subgroups isomorphic to Zp , hence is L semi-simple. Then B is a bounded pure subgroup of Cp , therefore Cp = B D1 for some divisible subgroup D1 of Cp . L But Cp is reduced hence D1 = 0 and also Cp is semisimple. L Then ∼ C = Cp (where p is prime) is also semisimple. Now if C M N with = L ∼ ∼ M = Zp and D = K L with K = Zp , then there is a monomorphism g : M −→L K with Img for the endomorphism f : G −→ G L LK. Therefore L of G = D C = K L M N defined by f (k, l, m, n) = (g(m), 0, 0, 0).
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L L L L L L We have Imf = Img 0 0 0 K L M N and f 6= 0 that is a contradiction with T -noncosingularity of G. So if C has a direct summand isomorphic to Zp for some prime p, then D has no direct summand isomorphic to Zp . (←−) If the conditions are satisfied then D is T -noncosingular related to D and C and also C Lis T -noncosingular related to D and C therefore by proposition 0.12 G = D C is T - noncosingular. Proposition 0.17. If a reduced group G is T -noncosingular, then T (G) is semisimple. Proof: this flows from Proposition 0.16 Proposition 0.18. For a torsion-free group G, If Radk+1 G E Radk G for k = 0, 1..., n−1, where Rad0 G = G and Radn G = 0 then G is T -noncosingular. Proof: let f : G −→ G be an endomorphism with Imf G. then f (G) = Imf ≤ RadG and f (RadG) ≤ Radf (G) ≤ Rad(RadG) = Rad2 G. Similarly f (Rad2 G) ≤ Radf (RadG) ≤ Rad(Rad2 G) = Rad3 G. Continuing in this way we will get f (Rad( n − 1)G) ≤ Radn G = 0. Since Radn−1 G E Radn−2 G for every 0 6= aRadn−2 G, we have 0 6= maRadn−1 G, for some mZ, therefore mf (a) = f (ma) = 0. since G is torsion-free, f (a) = 0.So f (Radn−2 G) = 0. Continuing tin this way we get f (G) = 0, that is f = 0 , hence G is T noncosingular. a |(m, p) = 1}, then Bp is not T -noncosingular.(Bp Example 0.19. Let Bp = { m is a lacal group) Proof: Note that Bp is torsion-free group and pBp is the largest subgroup of Bp but pBp Bp . Now take a non zero endomorphism f : Bp −→ Bp defined by f (x) = px, then Imf ≤ pBp Bp . Therefore Imf Bp and hence Bp is not T -noncosingular.
4. CONCLUSIONS For a torsion group, we are able to fully characterized the notion of T noncosingular abelian group and precisely stated that, for a torsion group to be T -noncosingular the group G, must be decomposed as G = D ⊕ C, where D is divisible and C is semi-simple. Also we established that, for a torsion T -noncosingular group G with Imf * RadG for all nonzero f produces semisimple group G. Our characterization for torsion group is surprisingly working; but the situation of torsion-free group is still subject to further research; however we provide a result that gives more information of the notion for every abelian group G.
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Acknowledgements. The author is very grateful to Prof Dr Rafail Alizade for his valuable contributions and suggestion towards improving this work, the author is highly grateful to to his Excellency the executive Governor of Kano state for his support and sponsorship.
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[15] W.J. Wickless, A First Graduate Course in Abstract Algebra, Marcel Dekker Inc, 2004. [16] R. Wisbauer, Foundations of Module and Ring Theory. Reading: Gordon and Breach Science Publishers, 1991. Received: January 17, 2015; Published: April 4, 2015
