Additive unit structure of endomorphism rings and invariance of modules
Tóm tắt
We use the type theory for rings of operators due to Kaplansky to describe the structure of modules that are invariant under automorphisms of their injective envelopes. Also, we highlight the importance of Boolean rings in the study of such modules. As a consequence of this approach, we are able to further the study initiated by Dickson and Fuller regarding when a module invariant under automorphisms of its injective envelope is invariant under any endomorphism of it. In particular, we find conditions for several classes of noetherian rings which ensure that modules invariant under automorphisms of their injective envelopes are quasi-injective. In the case of a commutative noetherian ring, we show that any automorphism-invariant module is quasi-injective. We also provide multiple examples to show that our conditions are the best possible, in the sense that if we relax them further then there exist automorphism-invariant modules which are not quasi-injective. We finish this paper by dualizing our results to the automorphism-coinvariant case.
Tài liệu tham khảo
Alahmadi, A., Facchini, A., Tung, N.K.: Automorphism-invariant modules. Rend. Sem. Mat. Univ. Padova 133, 241–259 (2015)
Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Graduate Texts in Mathematics, vol. 13. Springer, New York (1992)
Breaz, S., Călugăreanu, G., Schultz, P.: Subgroups which admit extensions of homomorphisms. Forum Math. 27(5), 2533–2549 (2013)
Clark, J., Lomp, C., Vanaja, N., Wisbauer, R.: Lifting Modules: Supplements and Projectivity in Module Theory. Frontiers in Mathematics. Birkhauser Verlag, Basel (2006)
Dickson, S.E., Fuller, K.R.: Algebras for which every indecomposable right module is invariant in its injective envelope. Pac. J. Math. 31(3), 655–658 (1969)
Dieudonné, J.: La théorie de Galois des anneux simples et semi-simples. Comment. Math. Helv. 21, 154–184 (1948)
Er, N., Singh, S., Srivastava, A.K.: Rings and modules which are stable under automorphisms of their injective hulls. J. Algebra 379, 223–229 (2013)
Facchini, A.: Module Theory. Endomorphism rings and direct sum decompositions in some classes of modules, Progress in Math. 167, Birkhäuser Verlag, 1998. Reprinted in Modern Birkhäuser Classics, Birkhäuser Verlag (2010)
Goldsmith, B., Pabst, S., Scott, A.: Unit sum numbers of rings and modules. Quart. J. Math. Oxf. (2) 49, 331–344 (1998)
Goodearl, K.R., Warfield, R.B.: An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts 16. Cambridge University Press, Cambridge (1989)
Guil Asensio, P.A., Keskin Tütüncü, D., Srivastava, A.K.: Modules invariant under automorphisms of their covers and envelopes. Isr. J. Math 206(1), 457–482 (2015)
Guil Asensio, P.A., Keskin Tütüncü, D., Kalegobaz, B., Srivastava, A.K.: Modules coinvariant under automorphisms of their projective covers. J. Algebra 466, 147–152 (2016)
Guil Asensio, P.A., Srivastava, A.K.: Automorphism-invariant modules satisfy the exchange property. J. Algebra 388, 101–106 (2013)
Guil Asensio, P.A., Srivastava, A.K.: Additive unit representations in endomorphism rings and an extension of a result of Dickson and Fuller. Ring theory and its applications, contemporary mathematics. Am. Math. Soc 609, 117–121 (2014)
Guil Asensio, P.A., Srivastava, A.K.: Automorphism-invariant modules. Noncommutative rings and their applications, Contemp. Math., Amer. Math. Soc 634, 19–30 (2015)
Jategaonkar, A.V.: Jacobson’s conjecture and modules over fully bounded noetherian rings. J. Algebra 30, 103–121 (1974)
Jain, S.K., Singh, S.: Quasi-injective and pseudo-injective modules. Can. Math. Bull. 18, 359–366 (1975)
Johnson, R.E., Wong, F.T.: Quasi-injective modules and irreducible rings. J. Lond. Math. Soc. 36, 260–268 (1961)
Kaplansky, I.: Rings of Operators. Benjamin, New York (1968)
Khurana, D., Srivastava, A.K.: Right self-injective rings in which each element is sum of two units. J. Algebra Appl. 6(2), 281–286 (2007)
Khurana, D., Srivastava, A.K.: Unit sum numbers of right self-injective rings. Bull. Austr. Math. Soc. 75(3), 355–360 (2007)
Lee, T.K., Zhou, Y.: Modules which are invariant under automorphisms of their injective hulls. J. Algebra Appl. 12(2), 1250159 (2013)
Mohamed, S.H., Müller, B. J.: Continuous and Discrete Modules. London Math. Soc. LN 147: Cambridge University Press, Cambridge (1990)
Murray, F.J., von Neumann, J.: On rings of operators. Ann. Math. 36, 116–229 (1937)
Nicholson, W.K., Zhou, Y.: Clean general rings. J. Algebra 291, 297–311 (2005)
Singh, S., Srivastava, A.K.: Rings of invariant module type and automorphism-invariant modules. Ring theory and its applications, contemporary mathematics. Am. Math. Soc. 609, 299–311 (2014)
Stenström, B.: Rings and Modules of Quotients. Lecture notes in mathematics, vol. 237. Springer, Berlin (1971)
Teply, M.L.: Pseudo-injective modules which are not quasi-injective. Proc. Am. Math. Soc. 49(2), 305–310 (1975)
Vámos, P.: 2-Good rings. Quart. J. Math. 56, 417–430 (2005)
Vámos, P., Wiegand, S.: Block diagonalization and 2-unit sums of matrices over prüfer domains. Trans. Am. Math. Soc. 363, 4997–5020 (2011)
Wolfson, K.G.: An ideal theoretic characterization of the ring of all linear transformations. Am. J. Math. 75, 358–386 (1953)
Zelinsky, D.: Every linear transformation is sum of nonsingular ones. Proc. Am. Math. Soc. 5, 627–630 (1954)