Research Article
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Year 2018, , 91 - 106, 05.07.2018
https://doi.org/10.24330/ieja.440231

Abstract

References

  • M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Alge- bra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
  • K. Brown, P. A. A. B. Carvalho and J. Matczuk, Simple modules and their essential extensions for skew polynomial rings, ArXiv e-prints, (2017), available at 1705.06596.
  • P. A. A. B. Carvalho, C. Lomp and D. Pusat-Yilmaz, Injective modules over down-up algebras, Glasg. Math. J., 52(A) (2010), 53-59.
  • P. A. A. B. Carvalho and I. M. Musson, Monolithic modules over Noetherian rings, Glasg. Math. J., 53(3) (2011), 683-692.
  • P. A. A. B. Carvalho, C. Hatipoglu and C. Lomp, Injective hulls of simple modules over di erential operator rings, Comm. Algebra, 43(10) (2015), 4221- 4230.
  • G. Cauchon, Anneaux de polyn^omes essentiellement bornes, Ring theory (Proc. Antwerp Conf. (NATO Adv. Study Inst.), Univ. Antwerp, Antwerp, 1978), Lecture Notes in Pure and Appl. Math., vol. 51, Dekker, New York, (1979), 27-42.
  • I. S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J., 17 (1950), 27-42.
  • C. Hatipoglu, Stable torsion theories and the injective hulls of simple modules, Int. Electron. J. Algebra, 16 (2014), 89-98.
  • C. Hatipoglu and C. Lomp, Injective hulls of simple modules over nite di- mensional nilpotent complex Lie superalgebras, J. Algebra, 361 (2012), 79-91.
  • D. Hilbert, Ein Beitrag zur Theorie des Legendre'schen Polynoms, Acta Math., 18(1) (1894), 155-159.
  • J. P. Jans, On co-Noetherian rings, J. London Math. Soc., 1(2) (1969), 588-590.
  • A. V. Jategaonkar, Certain injectives are Artinian, Noncommutative ring the- ory (Internat. Conf., Kent State Univ., Kent Ohio, 1975), Lecture Notes in Math., Vol. 545, Springer, Berlin, (1976), 128-139.
  • E. Matlis, Injective modules over Noetherian rings, Paci c J. Math., 8 (1958), 511-528.
  • I. M. Musson, Finitely generated, non-Artinian monolithic modules, New trends in noncommutative algebra, Contemp. Math., 562, Amer. Math. Soc., Providence, RI, (2012), 211-220.
  • A. Rosenberg and D. Zelinsky, Finiteness of the injective hull, Math. Z., 70 (1958/1959), 372-380.
  • A. Sant'Ana and R. Vinciguerra, On cyclic essential extensions of simple modules over di erential operator rings, ArXiv e-prints, (2017), available at 1704.04970.
  • W. Schelter, Essential extensions and intersection theorems, Proc. Amer. Math. Soc., 53(2) (1975), 328-330.
  • D. W. Sharpe and P. Vamos, Injective Modules, Cambridge Tracts in Math- ematics and Mathematical Physics, No. 62, Cambridge University Press, London-New York, 1972.
  • P. F. Smith, The Artin-Rees property, Paul Dubreil and Marie-Paule Malli- avin Algebra Seminar, 34th Year (Paris, 1981), Lecture Notes in Math., 924, Springer, Berlin-New York, (1982), 197-240.
  • P. Vamos, The dual of the notion of \ nitely generated", J. London Math. Soc., 43 (1968), 643-646.

A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS

Year 2018, , 91 - 106, 05.07.2018
https://doi.org/10.24330/ieja.440231

Abstract

Matlis showed that the injective hull of a simple module over
a commutative Noetherian ring is Artinian. In several recent papers, non-
commutative Noetherian rings whose injective hulls of simple modules are lo-
cally Artinian have been studied. This property had been denoted by property
(). In this paper we investigate, which non-Noetherian semiprimary commu-
tative quasi-local rings (R;m) satisfy property (). For quasi-local rings (R;m)
with m3 = 0, we prove a characterization of this property in terms of the dual
space of Soc(R). Furthermore, we show that (R;m) satises () if and only if
its associated graded ring gr(R) does.
Given a eld F and vector spaces V and W and a symmetric bilinear
map : V V ! W we consider commutative quasi-local rings of the form
F V W, whose product is given by
(1; v1;w1)(2; v2;w2) = (12; 1v2 + 2v1; 1w2 + 2w1 + (v1; v2))
in order to build new examples and to illustrate our theory. In particular we
prove that a quasi-local commutative ring with radical cube-zero does not sat-
isfy () if and only if it has a factor, whose associated graded ring is of the
form F V F with V innite dimensional and non-degenerated.

References

  • M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Alge- bra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
  • K. Brown, P. A. A. B. Carvalho and J. Matczuk, Simple modules and their essential extensions for skew polynomial rings, ArXiv e-prints, (2017), available at 1705.06596.
  • P. A. A. B. Carvalho, C. Lomp and D. Pusat-Yilmaz, Injective modules over down-up algebras, Glasg. Math. J., 52(A) (2010), 53-59.
  • P. A. A. B. Carvalho and I. M. Musson, Monolithic modules over Noetherian rings, Glasg. Math. J., 53(3) (2011), 683-692.
  • P. A. A. B. Carvalho, C. Hatipoglu and C. Lomp, Injective hulls of simple modules over di erential operator rings, Comm. Algebra, 43(10) (2015), 4221- 4230.
  • G. Cauchon, Anneaux de polyn^omes essentiellement bornes, Ring theory (Proc. Antwerp Conf. (NATO Adv. Study Inst.), Univ. Antwerp, Antwerp, 1978), Lecture Notes in Pure and Appl. Math., vol. 51, Dekker, New York, (1979), 27-42.
  • I. S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J., 17 (1950), 27-42.
  • C. Hatipoglu, Stable torsion theories and the injective hulls of simple modules, Int. Electron. J. Algebra, 16 (2014), 89-98.
  • C. Hatipoglu and C. Lomp, Injective hulls of simple modules over nite di- mensional nilpotent complex Lie superalgebras, J. Algebra, 361 (2012), 79-91.
  • D. Hilbert, Ein Beitrag zur Theorie des Legendre'schen Polynoms, Acta Math., 18(1) (1894), 155-159.
  • J. P. Jans, On co-Noetherian rings, J. London Math. Soc., 1(2) (1969), 588-590.
  • A. V. Jategaonkar, Certain injectives are Artinian, Noncommutative ring the- ory (Internat. Conf., Kent State Univ., Kent Ohio, 1975), Lecture Notes in Math., Vol. 545, Springer, Berlin, (1976), 128-139.
  • E. Matlis, Injective modules over Noetherian rings, Paci c J. Math., 8 (1958), 511-528.
  • I. M. Musson, Finitely generated, non-Artinian monolithic modules, New trends in noncommutative algebra, Contemp. Math., 562, Amer. Math. Soc., Providence, RI, (2012), 211-220.
  • A. Rosenberg and D. Zelinsky, Finiteness of the injective hull, Math. Z., 70 (1958/1959), 372-380.
  • A. Sant'Ana and R. Vinciguerra, On cyclic essential extensions of simple modules over di erential operator rings, ArXiv e-prints, (2017), available at 1704.04970.
  • W. Schelter, Essential extensions and intersection theorems, Proc. Amer. Math. Soc., 53(2) (1975), 328-330.
  • D. W. Sharpe and P. Vamos, Injective Modules, Cambridge Tracts in Math- ematics and Mathematical Physics, No. 62, Cambridge University Press, London-New York, 1972.
  • P. F. Smith, The Artin-Rees property, Paul Dubreil and Marie-Paule Malli- avin Algebra Seminar, 34th Year (Paris, 1981), Lecture Notes in Math., 924, Springer, Berlin-New York, (1982), 197-240.
  • P. Vamos, The dual of the notion of \ nitely generated", J. London Math. Soc., 43 (1968), 643-646.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Paula A. A. B. Carvalho, This is me

Christian Lomp

Patrick F. Smith

Publication Date July 5, 2018
Published in Issue Year 2018

Cite

APA Carvalho, P. A. A. B., Lomp, C., & Smith, P. F. (2018). A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS. International Electronic Journal of Algebra, 24(24), 91-106. https://doi.org/10.24330/ieja.440231
AMA Carvalho, PAAB, Lomp C, Smith PF. A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS. IEJA. July 2018;24(24):91-106. doi:10.24330/ieja.440231
Chicago Carvalho, Paula A. A. B., Christian Lomp, and Patrick F. Smith. “A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS”. International Electronic Journal of Algebra 24, no. 24 (July 2018): 91-106. https://doi.org/10.24330/ieja.440231.
EndNote Carvalho, PAAB, Lomp C, Smith PF (July 1, 2018) A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS. International Electronic Journal of Algebra 24 24 91–106.
IEEE P. A. A. B. Carvalho, C. Lomp, and P. F. Smith, “A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS”, IEJA, vol. 24, no. 24, pp. 91–106, 2018, doi: 10.24330/ieja.440231.
ISNAD Carvalho,, Paula A. A. B. et al. “A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS”. International Electronic Journal of Algebra 24/24 (July 2018), 91-106. https://doi.org/10.24330/ieja.440231.
JAMA Carvalho, PAAB, Lomp C, Smith PF. A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS. IEJA. 2018;24:91–106.
MLA Carvalho, Paula A. A. B. et al. “A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS”. International Electronic Journal of Algebra, vol. 24, no. 24, 2018, pp. 91-106, doi:10.24330/ieja.440231.
Vancouver Carvalho, PAAB, Lomp C, Smith PF. A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS. IEJA. 2018;24(24):91-106.

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