Local morphisms and modules with a semilocal endomorphism ring. (English) Zbl 1130.16014
A semilocal ring is a ring whose factor by the Jacobson radical is semisimple. A homomorphism of rings is local if every element with invertible image is itself invertible.
Based on the fact that every ring which has a local homomorphism into a semilocal ring is itself semilocal, the article gives several examples for modules having a semilocal endomorphism ring, such as (1) finitely presented modules over semilocal rings (Theorem 3.3), (2) finitely copresented modules, i.e., modules having an injective extension of finite Goldie dimension by a module of finite Goldie dimension (Corollary 5.5), (3) torsion-free finite-rank modules over an algebra over a commutative Noetherian semilocal domain of Krull dimension 1 (Corollary 5.9), (4) modules having both finite Goldie dimension and finite dual Goldie dimension (Corollary 6.5).
Some results are generalized to Grothendieck categories. Along the way, a local homomorphism from every ring into a von Neumann regular ring is constructed (Theorem 5.3).
The essential part of the proofs is the construction of local homomorphisms. These are mainly homomorphisms between endomorphism rings induced by additive functors. Notable additive functors are the canonical functor to the spectral category (obtained by making all essential monomorphisms invertible) and the dual one.
Based on the fact that every ring which has a local homomorphism into a semilocal ring is itself semilocal, the article gives several examples for modules having a semilocal endomorphism ring, such as (1) finitely presented modules over semilocal rings (Theorem 3.3), (2) finitely copresented modules, i.e., modules having an injective extension of finite Goldie dimension by a module of finite Goldie dimension (Corollary 5.5), (3) torsion-free finite-rank modules over an algebra over a commutative Noetherian semilocal domain of Krull dimension 1 (Corollary 5.9), (4) modules having both finite Goldie dimension and finite dual Goldie dimension (Corollary 6.5).
Some results are generalized to Grothendieck categories. Along the way, a local homomorphism from every ring into a von Neumann regular ring is constructed (Theorem 5.3).
The essential part of the proofs is the construction of local homomorphisms. These are mainly homomorphisms between endomorphism rings induced by additive functors. Notable additive functors are the canonical functor to the spectral category (obtained by making all essential monomorphisms invertible) and the dual one.
Reviewer: Gábor Braun (Budapest)
MSC:
| 16L30 | Noncommutative local and semilocal rings, perfect rings |
| 16W20 | Automorphisms and endomorphisms |
| 16S50 | Endomorphism rings; matrix rings |
| 18E15 | Grothendieck categories (MSC2010) |
| 16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |
| 18E35 | Localization of categories, calculus of fractions |
Keywords:
semilocal rings; local homomorphisms; endomorphism rings; spectral categories; Goldie dimension; Krull dimension; Grothendieck categories; von Neumann regular ringsReferences:
| [1] | Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules, 2nd Edition. Springer, Berlin Heidelberg New York (1992) · Zbl 0765.16001 |
| [2] | Björk, J.-E.: Conditions which imply that subrings of artinian rings are artinian. J. Reine Angew. Math. 247, 123–138 (1971) · Zbl 0211.36402 · doi:10.1515/crll.1971.247.123 |
| [3] | Björk, J.-E.: Conditions which imply that subrings of semiprimary rings are semiprimary. J. Algebra 19, 384–395 (1971) · Zbl 0221.16019 · doi:10.1016/0021-8693(71)90097-4 |
| [4] | Camps, R., Dicks, W.: On semilocal rings. Israel J. Math. 81, 203–211 (1993) · Zbl 0802.16010 · doi:10.1007/BF02761306 |
| [5] | Camps, R., Menal, P.: Power cancellation for artinian modules. Comm. Algebra 19, 2081–2095 (1991) · Zbl 0732.16016 · doi:10.1080/00927879108824246 |
| [6] | Cohn, P.M.: Skew Field Constructions. London Math. Soc. Lecture Notes Ser. 27. Cambridge University Press (1977) · Zbl 0355.16009 |
| [7] | Diracca, L., Facchini, A.: Uniqueness of monogeny classes for uniform objects in abelian categories. J. Pure Appl. Algebra 172, 183–191 (2002) · Zbl 1006.18010 · doi:10.1016/S0022-4049(01)00160-8 |
| [8] | Facchini, A.: Module theory. Endomorphism rings and direct sum decompositions in some classes of modules, Progress in Math. vol. 167. Birkhäuser Verlag, Basel (1998) · Zbl 0930.16001 |
| [9] | Facchini, A.: Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids. J. Algebra 256(1), 280–307 (2002) · Zbl 1016.16002 · doi:10.1016/S0021-8693(02)00164-3 |
| [10] | Facchini, A., Herbera, D.: Projective modules over semilocal rings. In: Huynh, D.V., Jain, S.K., López-Permouth, S.R. (eds.) Algebra and its Applications, Contemporary Math. vol. 259, Amer. Math. Soc., Providence, pp. 181–198 (2000) · Zbl 0981.16003 |
| [11] | Gabriel, P., Oberst, U.: Spektralkategorien und reguläre Ringe im Von-Neumannschen Sinn. Math. Zeitschr. 82, 389–395 (1966) · Zbl 0136.25602 · doi:10.1007/BF01112218 |
| [12] | Herbera, D., Shamsuddin, A.: Modules with semi-local endomorphism ring. Proc. Amer. Math. Soc. 123, 3593–3600 (1995) · Zbl 0843.16017 · doi:10.1090/S0002-9939-1995-1277114-8 |
| [13] | Jensen, C.U., Lenzing, H.: Model Theoretic Algebra. Algebra, Logic and Applications Series, vol. 2. Gordon and Breach Science Publishers, New York (1989) · Zbl 0728.03026 |
| [14] | Matlis, E.: Some properties of noetherian domains of dimension one. Canad. J. Math. 13, 569–586 (1961) · Zbl 0102.02901 · doi:10.4153/CJM-1961-046-x |
| [15] | Popescu, N.: Abelian Categories with Applications to Rings and Modules. Academic Press, London New York (1973) · Zbl 0271.18006 |
| [16] | Rowen, L.H.: Finitely presented modules over semiperfect rings. Proc. Amer. Math. Soc. 97, 1–7 (1986) · Zbl 0596.16020 · doi:10.1090/S0002-9939-1986-0831374-8 |
| [17] | Schofield, A.H.: Representations of rings over skew fields. London Math. Soc. Lecture Notes Series, vol. 92. Cambridge University Press (1985) · Zbl 0571.16001 |
| [18] | Stenström, B.: Rings of Quotiens. Springer, Berlin Heidelberg New York (1975) · Zbl 0296.16001 |
| [19] | Vasconcelos, W.V.: On finitely generated flat modules. Trans. Amer. Math. Soc. 138, 505–512 (1969) · Zbl 0175.03603 · doi:10.1090/S0002-9947-1969-0238839-5 |
| [20] | Vasconcelos, W.V.: Injective endomorphisms of finitely generated flat modules. Proc. Amer. Math. Soc. 25, 900–901 (1970) · Zbl 0197.31404 |
| [21] | Warfield, R.B. Jr.: Cancellation of modules and groups and stable range of endomorphism rings. Pacific J. Math. 91(2), 457–485 (1980) · Zbl 0484.16017 |
| [22] | Wiegand, R.: Direct-sum decompositions over local rings. J. Algebra 240, 83–97 (2001) · Zbl 0986.13010 · doi:10.1006/jabr.2000.8657 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
