Abstract
This paper investigates situations where a property of a ring can be tested on a set of “prime right ideals.” Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every “prime right ideal” is finitely generated (resp. principal), where the phrase “prime right ideal” can be interpreted in one of many different ways. We also use our methods to show that other properties can be tested on special sets of right ideals, such as the right artinian property and various homological properties. Applying these methods, we prove the following noncommutative generalization of a result of Kaplansky: a (left and right) noetherian ring is a principal right ideal ring iff all of its maximal right ideals are principal. A counterexample shows that the left noetherian hypothesis cannot be dropped. Finally, we compare our results to earlier generalizations of Cohen’s and Kaplansky’s theorems in the literature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Amitsur, S.A., Small, L.W.: Finite-dimensional representations of PI algebras. J. Algebra 133(2), 244–248 (1990). MR 1067405 (91h:16029)
Beachy, J.A., Weakley, W.D.: A note on prime ideals which test injectivity. Commun. Algebra 15(3), 471–478 (1987). MR 882795 (88f:16026)
Bhatwadekar, S.M.: On the global dimension of some filtered algebras. J. Lond. Math. Soc. (2) 13(2), 239–248 (1976) MR 0404398 (53 #8200)
Chandran, V.R.: On two analogues of Cohen’s theorem. Indian J. Pure Appl. Math. 8(1), 54–59 (1977) MR 0453809 (56 #12062)
Chandran, V.R.: On two analogues of Cohen’s theorem. Pure Appl. Math. Sci. 7(1–2), 5–10 (1978) MR 0460378 (57 #372)
Cohen, I.S.: Commutative rings with restricted minimum condition. Duke Math. J. 17, 27–42 (1950) MR 0033276 (11,413g)
Cohn, P.M.: Free Ideal Rings and Localization in General Rings. New Mathematical Monographs, vol. 3. Cambridge University Press, Cambridge (2006) MR 2246388 (2007k:16020)
Eisenbud, D.: Commutative Algebra: With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer-Verlag, New York (1995) MR 1322960 (97a:13001)
Evans, E.G. Jr.: Krull–Schmidt and cancellation over local rings. Pac. J.Math. 46, 115–121 (1973) MR 0323815 (48 #2170)
Goldie, A.W.: Non-commutative principal ideal rings. Arch. Math. 13, 213–221 (1962) MR 0140532 (25 #3951)
Goldie, A.W.: Properties of the idealiser. In: Ring Theory (Proc. Conf., Park City, Utah, 1971), pp. 161–169. Academic Press, New York (1972) MR 0382341 (52 #3226)
Goodearl, K.R.: Global dimension of differential operator rings. II. Trans. Am. Math. Soc. 209, 65–85 (1975) MR 0382359 (52 #3244)
Goodearl, K.R., Warfield, R.B. Jr.: An Introduction to Noncommutative Noetherian Rings. London Mathematical Society Student Texts, vol. 61, 2nd edn. Cambridge University Press, Cambridge (2004) MR 2080008 (2005b:16001)
Gordon, R., Robson, J.C.: Krull Dimension. American Mathematical Society, no. 133. Memoirs of the American Mathematical Society, Providence (1973) MR 0352177 (50 #4664)
Huynh, D.V.: A note on rings with chain conditions. Acta Math. Hungar. 51(1–2), 65–70 (1988) MR 934584 (89e:16024)
Jategaonkar, A.V.: Left principal ideal domains. J. Algebra 8, 148–155 (1968) MR 0218387 (36 #1474)
Jategaonkar, A.V.: A counter-example in ring theory and homological algebra. J. Algebra 12, 418–440 (1969) MR 0240131 (39 #1485)
Kaplansky, I.: Elementary divisors and modules. Trans. Am. Math. Soc. 66, 464–491 (1949) MR 0031470 (11,155b)
Kertész, A.: Noethersche ringe, die artinsch sind. Acta Sci. Math. (Szeged) 31, 219–221 (1970) MR 0279126 (43 #4852)
Koh, K.: On one sided ideals of a prime type. Proc. Am. Math. Soc. 28, 321–329 (1971) MR 0274488 (43 #251)
Koh, K.: On prime one-sided ideals. Can. Math. Bull. 14, 259–260 (1971) MR 0313325 (47 #1880)
Koker, J.J.: Global dimension of rings with Krull dimension. Commun. Algebra 20(10), 2863–2876 (1992) MR 1179266 (94a:16011)
Krause, G.: On fully left bounded left noetherian rings. J. Algebra 23, 88–99 (1972) MR 0308188 (46 #7303)
Lam, T.Y.: Lectures on Modules and Rings. Graduate Texts in Mathematics, vol. 189. Springer-Verlag, New York (1999) MR 1653294 (99i:16001)
Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Mathematics, vol. 131, 2nd edn. Springer-Verlag, New York (2001) MR 1838439 (2002c:16001)
Lam, T.Y.: Exercises in Classical Ring Theory. Problem Books in Mathematics, 2nd edn. Springer-Verlag, New York (2003) MR 2003255 (2004g:16001)
Lam, T.Y.: A crash course on stable range, cancellation, substitution and exchange. J. Algebra Appl. 3(3), 301–343 (2004) MR 2096452 (2005g:16007)
Lam, T.Y.: Exercises in Modules and Rings. Problem Books in Mathematics. Springer, New York (2007) MR 2278849 (2007h:16001)
Lam, T.Y., Reyes, M.L.: A Prime Ideal Principle in commutative algebra. J. Algebra 319(7), 3006–3027 (2008) MR 2397420 (2009c:13003)
McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Graduate Studies in Mathematics, vol. 30, revised ed. American Mathematical Society, Providence (2001) MR 1811901 (2001i:16039)
Michler, G.O.: Prime right ideals and right noetherian rings. In: Ring Theory (Proc. Conf., Park City, Utah, 1971), pp. 251–255. Academic Press, New York (1972) MR 0340334 (49 #5089)
Osofsky, B.L.: A generalization of quasi-Frobenius rings. J. Algebra 4, 373–387 (1966) MR 0204463 (34 #4305)
Reyes,M.L.: A one-sided Prime Ideal Principle for noncommutative rings. J. Algebra Appl. 9(6), 877–919 (2010)
Robson, J.C.: Rings in which finitely generated right ideals are principal. Proc. Lond. Math. Soc. 17(3), 617–628 (1967) MR 0217109 (36 #200)
Robson, J.C.: Decomposition of noetherian rings. Commun. Algebra 1, 345–349 (1974) MR 0342564 (49 #7310)
Rotman, J.J.: An Introduction to Homological Algebra. Pure and Applied Mathematics, vol. 85. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1979) MR 538169 (80k:18001)
Smith, P.F.: Injective modules and prime ideals. Commun. Algebra 9(9), 989–999 (1981) MR 614468 (82h:16018)
Smith, P.F.: The injective test lemma in fully bounded rings. Commun. Algebra 9(17), 1701–1708 (1981) MR 631883 (82k:16030)
Smith, P.F.: Concerning a theorem of I. S. Cohen. An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 2, 160–167 (1994) XIth National Conference of Algebra (Constanţa, 1994). MR 1367558 (96m:16029)
Zabavs’kiĭ, B.V.: A noncommutative analogue of Cohen’s theorem. Ukr. Mat. Z. 48(5), 707–710 (1996) MR 1417038
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was supported in part by a Ford Foundation Predoctoral Diversity Fellowship. This work is a portion of his Ph.D. thesis at the University of California, Berkeley.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Reyes, M.L. Noncommutative Generalizations of Theorems of Cohen and Kaplansky. Algebr Represent Theor 15, 933–975 (2012). https://doi.org/10.1007/s10468-011-9273-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-011-9273-7
Keywords
- Point annihilators
- Cocritical right ideals
- Cohen’s theorem
- Right noetherian rings
- Kaplansky’s theorem
- Principal right ideals