Abstract
This survey is based on the PhD Thesis that was defended at the Dissertation council of the Faculty of Mechanics and Mathematics of Moscow State University on December 6, 2013. This paper is devoted to the study of quotient rings of rings graded by a group. Graded analogs of the Faith–Utumi theorem of orders of matrix rings and Goldie’s theorems of orders of completely reducible rings are proved, and the orthogonal graded completion, which is an analog of the quotient ring underlying the orthogonal completion theory of Beidar–Mikhalev, is constructed.
Similar content being viewed by others
References
O. D. Avraamova, The Generalized Density Theorems, Candidate’s Dissertation in Physics and Mathematics, Moscow (1989).
I. N. Balaba, “Rings of quotients of semiprime graded rings,” in: Proc. Int. Seminar ”Universal Algebra and Its Applications,” Volgograd (2000), pp. 21–28.
I. N. Balaba, Graded Rings and Modules, Doctoral Dissertation in Physics and Mathematics, Moscow (2012).
I. N. Balaba and V. A. Efremov, “Graded rings of quotients of semiprime graded rings,” Chebyshevskii Sb., 11, No. 1 (33), 20–30 (2010).
I. N. Balaba, A. L. Kanunnikov, and A. V. Mikhalev, “Graded quotient ring of associative rings. I,” Fundament. Prikl. Mat., 17, No. 2, 3–74 (2012).
K. I. Beidar, “Rings with generalized identities. I,” Moscow Univ. Math. Bull., 32, No. 1, 15–20 (1977).
K. I. Beidar, “Rings of quotients of semiprime rings,” Moscow Univ. Math. Bull., 33, No. 5, 29–34 (1978).
K. I. Beidar, W. S. Martindale, and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, New York (1995).
K. I. Beidar and A. V. Mikhalev, “Orthogonal completeness and algebraic systems,” Russ. Math. Surv., 40, No. 6, 51–95 (1985).
K. I. Beidar and A. V. Mikhalev, “Functor of orthogonal completion,” Abelian Groups Modules, No. 4, 3–19 (1986).
Chen-Lian Chuang, “Boolean valued models and semiprime rings,” in: Rings and Nearrings. Proc. Int. Conf. Algebra in Memory of Kostia Beidar, Walter de Gruyter, Berlin (2005), pp. 23–53.
C. Faith, Algebra: Rings, Modules and Categories, Vol. 1, Springer, Berlin (1973).
A. W. Goldie, “The structure of prime rings under ascending chain conditions,” Proc. London Math. Soc., 8, 589–608 (1958).
A. W. Goldie, “Semi-prime rings with maximal conditions,” Proc. London Math. Soc., 10, 201–220 (1960).
A. W. Goldie, “Non-commutative principal ideal rings,” Arch. Math., 13, 213-221 (1962).
K. Goodearl and T. Stafford, “The graded version of Goldie’s theorem,” in: Algebra and Its Applications. Int. Conf. Algebra and Its Applications, March 25–28, 1999, Ohio Univ., Athens, Contemp. Math., Vol. 259, Amer. Math. Soc., Providence (2000), pp. 237–240.
I. N. Herstein, Noncommutative Rings, Carus Math. Monographs, Vol. 15, Math. Assoc. America, Washington (1968).
I. N. Herstein, “A note on derivations,” Can. Math. Bull., 21, 369–370 (1978).
N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Publ., Vol. 37, Amer. Math. Soc., Providence (1964).
A. V. Jatengaonkar, Left Principal Ideal Rings, Lect. Notes Math., Vol. 123, Springer, Berlin (1970).
E. Jespers and P. Wauters, “A general notion of noncommutative Krull rings,” J. Algebra, 112, 388–415 (1988).
A. L. Kanunnikov, “Graded versions of Goldie’s theorem,” Moscow Univ. Math. Bull., 66, No. 3, 119–122 (2011).
A. L. Kanunnikov, “An application of the method of orthogonal completeness in graded ring theory,” Algebra Logic, 52, No. 4, 98–104 (2013).
A. L. Kanunnikov, “Graded versions of Goldie’s theorem. II,” Moscow Univ. Math. Bull., 68, No. 3, 162–165 (2013).
A. L. Kanunnikov, “Orders of graded matrix rings,” Vestn. MGADA, No. 1 (20), 52–56 (2013).
A. L. Kanunnikov, “Orthogonal graded completion of graded semiprime rings,” Fundam. Prikl. Mat., 17, No. 7, 117–150 (2011/12).
V. K. Kharchenko, Noncommutative Galois Theory [in Russian], Nauch. Kniga, Novosibirsk (1996).
J. Lambek, Lectures on Rings and Modules, Blaisdell, London (1966).
S. Liu, M. Beattie, and H. Fang, “Graded division rings and the Jacobson density theorem,” J. Beijing Norm. Univ. (Nat. Sci.), 27, No. 2, 129–134 (1991).
A. VMikhalev, “. Orthogonally full multisort systemsx,” Dokl. Akad. Nauk SSSR, 289, No. 6, 1304–1308.
C. Nᾰstᾰsescu, “Some construction over graded rings. Application,” J. Algebra, 120, 119–138 (1989).
C. Nᾰstᾰsescu, E. Nauwelaerts, and F. van Oystaeyen, “Arithmetically graded rings revisited,” Commun. Algebra, 14, No. 10, 1191–2017 (1986).
C. Nᾰstᾰsescu and F. van Oystaeyen, Graded and Filtered Rings and Modules, Lect. Notes Math., Vol. 758, Springer, Berlin (1979).
C. Nᾰstᾰsescu and F. van Oystaeyen, Graded Ring Theory, North-Holland, Amsterdam (1982).
C. Nᾰstᾰsescu and F. van Oystaeyen, Methods of Graded Rings, North-Holland, Amsterdam (2004).
K. Tewari, “Complexes over a complete algebra of quotients,” Can. J. Math., 19, 40–57 (1967).
A. A. Tuganbaev, Rings Theory. Arithmetical Modules and Rings, MCCME, Moscow (2009).
Y. Utumi, “On quotient rings,” Osaka J. Math., 8, 1–18 (1956).
H. Yahya, “A note on graded regular rings,” Commun. Algebra, 25, No. 1, 223–228 (1997).
V. K. Zakharov, “The orthogonal completion of rings and Boolean algebras,” Ordered Sets and Lattices, No. 4, 54–65 (1976).
A. E. Zalessky and A. V. Mikhalev, “Group rings,” J. Math. Sci., 4, No. 1, 1–78 (2005).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 20, No. 6, pp. 77–145, 2015.
Rights and permissions
About this article
Cite this article
Kanunnikov, A.L. Graded Quotient Rings. J Math Sci 233, 50–94 (2018). https://doi.org/10.1007/s10958-018-3925-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-018-3925-7