Projective modules with semilocal endomorphism ring

On projective modules with semilocal endomorphims ring. C. Lomp, M. Nasrutdinov and I. Sakhaev October 21, 2010 Abstract It is proved that a projective left R-module Pwith semilocal endomorphism ring is finitely generated if and only if hdim(P) = hdim((P/J(P)). 1 Preliminaries. Throughout Rdenotes an associative ring with unit and all modules will be left unitial R-module. Let J(M) be the Jacobson radical of an R-module M. A left R-module is called semi-local if M/J(M) is a semi-simple left R-module. For a semisimple module M=LiinI Miwhere all Miare simple R-modules, we denote by length (M) = card(I) the length of M. A submodule Nof Mis called small in Mif for every submodule Uof M with N+U=Mwe have U=M. We denote a small submodule Nof M by NM. A module Mis said to be hollow if M6= 0 and every proper submodule of Mis small in M.Mis said to have finite hollow dimension ( or finite dual Goldie dimension ) if there exists an exact sequence Mg −−−−→ Ln i=1 Hi−−−−→ 0 where all the Hiare hollow and the kernel of gis small in M. Then nis an invariant of the module Mand called the hollow dimension of Mand we write hdim (M) = n. In [SV, Theorem 1.8] it is shown that in this case Mdoes not allow any epimorphism to a direct sum with more then nsummands. Note that for a semisimple module Mwith finite dual Goldie dimensioin we have hdim (M) = length (M). The dual Goldie dimension was investigated in [GP], [HaS], [Lo], [SV], [T76], [T94], [V]. The modules with semilocal endomorphisms ring were investigated in [HeS], [Lo] and [Lo]. In [Lo] C.Lomp has formulated the question: Is every (self-) projective R-module Pwith semilocal endomorphisms ring finitely generated ? This question is closely related to an old problem of D. Lazard (see [La]): If Pis a projective left R-module with P/J(P)finitely generated then Pis finitely generated ? 1 H. Zoschinger in [Z81] and I.Sakhaev in [S77], [S85], [S87], [S89], [S91], [S93] have investigated the problem of D. Lazard. A non commutative semilocal ring over which there exists a no finitely generated projective module P such that P/J(P) is finitely generated was constructed by I. Sakhaev and V. Gerasimov in [GS]. 2 Proof of the Results. At first we show how our question is related to the problem of D. Lazard. Lemma 2.1. Let Pbe a projective left R-module with semilocal endomorphism ring. Then the left R-module P/J(P)is finitely generated. Proof. By Takeuchi’s Theorem ( see [Lo, Theorem 2.10] for a short proof) we have hdim (End (P)) = hdim (P)<∞and by [Lo, Theorem 2.7] Pis semilocal with hdim (P)≥length (P/J(P)). Since every semisimple module with finite length is finitely generated, the result follows. Now we will prove the more general confirmation about R-modules M with hdim (M)<∞. Theorem 2.2. Let M be a left R-module with hdim (M)finite. Then the following conditions are equivalent: (a) Mis finitely generated. (b) J(M)is small in M. (c) hdim (M)=hdim (M/J(M)). Proof. By [Lo, Theorem 2.7] Mis semilocal and hdim(M)≥length(M/J(M)). Hence M/J(M) is a semisimple module of finite length and hence finitely generated. By [W, 21.6(4)] a module Mis finitely generated if and only if M/J(M) is finitely generated and J(M)M. Hence (a)⇔(b) follows from [W, 21.6(4)]. As hdim (M) is finite (b)⇔(c) follows from [V, Theorem 1.20(3)]. Corollary 2.3. A projective left R-module P with semi-local endomorphisms ring is finitely generated if and only if hdim (P) = hdim (P/J(P)). Proof. By Takeuchi’s theorem [Lo, Theorem 3.10] we have hdim (P)<∞. Theorem 2.2 completes the proof. Remark 2.4. In [HeS, Example 10(2)] D. Herbera and A. Shamsuddin gave an example of a cyclic module with semilocal endomorphism ring, but with infinite dual Goldie dimension. Their construction is as follows: Take a ring extension R⊆Ssuch that Ris semilocal, but Sis not (for example R a field and S=R[X] the (commutative) polynomial ring with coefficients in R). Consider the (S, R)-bimodule M:= HomR(RS,RS/R), the sub-bimodule 2 N=f∈M|f(R)=0 and the ring T:= S M 0R. Take the right ideal I:= 0N 0Rof T. Then EndT(T/I)≃I0/I ≃Rholds as rings, where I0=R N 0Ris the idealizer of Iin T. Hence T/I is a cyclic T-module whose endomorphism ring is isomorphic to the semilocal ring R. Since Shas infinite dual Goldie dimension as right S-module, T/I has infinite dual Goldie dimension as right T-module. To see this let U:= 0M 0Rand note that SS=T/UTis a factor module of T/I as I⊆U. Thus ∞= hdim (S) = hdim (T/UT)≤hdim (T/I). References [SV] B.Sarath and K.Varadarajan, Dual Goldie dimension II, Commun. Algebra 7, 1885-1899 (1979). [GP] P.Grezeszcuk and E.R.Puczy lowski, On Goldie and dual Goldie dimension, J. Pure and Applied Algebra 31, 47-54 (1984) [HaS] A.Hanna and A.Shamsuddin, Dual Goldie dimension, Rendiconti dell’ Istitutio di Mathematica dell’ Universita di Trieste 24, No.1-2, 25-38 (1992) [Lo] C.Lomp, On dual Goldie dimension, Diplomarbeit, HHU D¨usseldorf (1996) [T76] T.Takeuchi, On cofinite-dimensional modules, Hokkaido Math. J. 5, 1-43 (1976) [T94] T.Takeuchi, Coranks of a quasi-projective module and its endomorphism ring, Glasgow Math. J. 36, 381 - 383 (1994) [V] K.Varadarajan, Dual Goldie dimension, Commun. Algebra, 7, 565-610 (1979) [HeS] D.Herbera and A.Shamsuddin, Modules with semi-local endomorphism ring, Proc. American Math. Soc. 123, 3593-3600 (1995) [Lo] C.Lomp, On semilocal modules and rings., Commun. Algebra 27, No.4, 1921-1935 (1999). [La] D. Lazard, Liberte des gros modules projectifs, J. Algebra 31, 437 - 451 (1974) [S77] I. Sakhaev, The finite generation of projective modules., Izv. VUZov. Mathematika. 9, 69 - 79 (1977) [S85] I. Sakhaev, On the projectivity of finitely generated flat modules over semilocal rings., Math. Notes 37 No. 2, 152 - 162 (1985) 3 [S87] I. Sakhaev, On the group K0(AP0).Math. Nachrichten 130, 157 - 175 (1987) [S89] I. Sakhaev, On SVI-rings and on D. Lazard hyphothesis. Math. Nachrichten 144, 87 - 98 (1989) [S91] I.I. Sakhaev, On lifting finite generacy of a projective module modulo its radical, Math. Notes 49, 293 - 301 (1991) [S93] I.I. Sakhaev, The finite generation of projective modules, Algebra(Proc. Int. Conf. Krasnoyarsk 1993), edited by Y.L. Ershov et al., 209-216, de Gruyter, Berlin 1996 [Z81] H.Z¨oschinger, Projektive Moduln mit endlich erzeugtem Radikalfaktormodul, Math. Annalen 255, 199 - 206 (1981) [GS] V.N. Gerasmiov and I.I. Sakhaev, A counterexample to two hypotheses on projective and flat modules, Siberian Math. J. 25, 855 - 859 (1984) [W] R.Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading (1991) 4