REFERENCES
H. Al-Ezeh, “Exchange PF-rings and almost PP-rings,” Mat. Vestnik 42, No. 2, 77-80 (1990).
P. Ara, “Strongly Π-regular rings have stable range one,” Proc. Amer. Math. Soc., 124, No. 11, 3293-3298 (1996).
P. Ara, “Extensions of exchange rings,” J. Algebra, 197, No. 2, 409-423 (1997).
P. Ara, M. Gómez-Lozano, and M. Siles Molina, “Local rings of exchange rings,” Commun. Algebra, 26, No. 12, 4191-4205 (1998).
P. Ara, K. C. O'Meara, and D. V. Tyukavkin, “Cancellation of projective modules over regular rings with comparability,” J. Pure Appl. Algebra, 107, No. 1, 19-38 (1996).
R. Arens and I. Kaplansky, “Topological representation of algebras,” Trans. Amer. Math. Soc., 63, 457-481 (1948).
E. P. Armendariz and J. W. Fisher, “Regular P.I.-rings,” Proc. Amer.Math. Soc., 39, No. 2, 247-251 (1973).
E. P. Armendariz, J. W. Fisher, and R. L. Snider, “On injective and surjective endomorphisms of definitely generated modules,” Commun. Algebra, 6, No. 7, 659-672 (1978).
E. P. Armendariz, J. W. Fisher, and S. A. Steinberg, “Central localization of regular rings,” Proc. Amer. Math. Soc., 46, No. 3, 315-321 (1974).
G. Azumaya, “Strongly Π-regular rings,” J. Fac. Sci. Hokkaido Univ., 13, 659-672 (1954).
G. Azumaya, “F-semiperfect modules,” J. Algebra, 136, 73-85 (1991).
G. Baccella, “On flat factor rings and fully right idempotent rings,” Ann. Univ. Ferrara, Sez. VII, 26, 125-141 (1980).
A. Badawi, “On semicommutative Π-regular rings,” Commun. Algebra, 22, No. 1, 151-157 (1993).
A. Badawi, “On Abelian Π-regular rings,” Commun. Algebra, 25, No. 4, 1009-1021 (1997).
K. I. Beidar and A. V. Mikhalev, “Semiprime rings with bounded index of nilpotent elements,” Tr. Sem. Petrovskogo, No. 13, 237-249 (1988).
K. I. Beidar and R. Wisbauer, “Strongly and properly semiprime rings and modules,” In: Ring Theory. Proc. Ohio State-Denison Conf. 1992, World Sci. Publ., Singapore (1993).
K. I. Beidar and R. Wisbauer, “Properly semiprime self-pp-modules,” Commun. Algebra 23, No. 3, 841-861 (1995).
L. P. Belluce, “Spectral spaces and noncommutative rings,” Commun. Algebra, 19, 1855-1865 (1991).
G. F. Birkenmeier, “A survey of regularity conditions and the simplicity of prime factor rings,” Vietnam J. Math., 23, No. 1, 29-38 (1995).
G. F. Birkenmeier, J. Y. Kim, and J. K. Park, “A connection between weak regularity and the simplicity of prime factor rings,” Proc. Amer. Math. Soc., 122, No. 1, 53-58 (1994).
G. F. Birkenmeier, J. Y. Kim, and J. K. Park, “Regularity conditions and the simplicity of prime factor rings,” J. Pure Appl. Algebra, 115, 213-230 (1997).
W. D. Blair and H. Tsutsui, “Fully prime rings,” Commun. Algebra, 22, No. 13, 5389-5400 (1994).
W. D. Burgess, “Confirmation of a conjecture of Dauns and Hofmann on biregular rings,” In: Abelian Groups and Modules(A. Facchini and C. Menini, Eds.), Kluwer Acad. Publ., Dordrecht (1995), pp. 67-72.
W. D. Burgess and P. Menal, “On strongly Π-regular rings and homomorphisms into them,” Commun. Algebra, 16, 1701-1725 (1988).
W. D. Burgess and W. Stephenson, “Pierce sheaves of noncommutative rings,” Commun. Algebra, 4, No. 1, 51-75 (1976).
W. D. Burgess and W. Stephenson, “An analogue of the Pierce sheaf for noncommutative rings,” Commun. Algebra, 6, No. 9, 863-886 (1978).
W. D. Burgess and W. Stephenson, “Rings all of whose Pierce stalks are local,” Can. Math. Bull., 22, No. 2, 159-164 (1979).
D. G. Burkholder, “Azumaya rings with locally perfect centers,” J. Algebra, 103, 606-618 (1986).
D. G. Burkholder, “Azumaya rings, Pierce stalks and central ideal algebras,” Commun. Algebra, 17, No. 1, 103-113 (1989).
D. G. Burkholder, “Products of Azumaya rings and Kochen's map,” Commun. Algebra, 17, No. 1, 115-134 (1989).
V. Camillo and Y. Xiao, “Weakly regular rings,” Commun. Algebra, 22, No. 10, 4095-4112 (1994).
V. Camillo and H.-P. Yu, “Exchange rings, units and idempotents,” Commun. Algebra, 22, No. 12, 4737-4749 (1994).
V. Camillo and H.-P. Yu, “Stable range one for rings with many idempotents,” Trans. Amer. Math. Soc., 347, No. 8, 3141-3147 (1995).
R. Camps and P. Menal, “Power cancellation for Artinian modules,” Commun. Algebra, 19, No. 7, 2081-2095 (1991).
R. Camps and P. Menal, “The power substitution property for rings of continuous functions,” J. Algebra, 161, No. 2, 455-466 (1993).
J. J. Carmona, J. Cufi, and P. Menal, “On the unit-1-stable rank of rings of analytic functions,” Publ. Mat., 26, No. 2A, 439-447 (1992).
H. Chen, “Exchange rings, related comparability and power substitution,” Commun. Algebra, 26, No. 10, 3383-3401 (1998).
M. Chacron, “On algebraic rings,” Bull. Austral. Math. Soc., 1, 385-389 (1969).
V. R. Chandran, “On two analogues of Cohen's theorem,” Indian J. Pure Appl. Math., 8, 54-59 (1977).
M. Contessa, “On PM-rings,” Commun. Algebra, 10, 93-108 (1982).
M. Contessa, “On certain classes of PM-rings,” Commun. Algebra, 12, No. 12, 1447-1469 (1984).
M. Contessa, “Ultraproducts of PM-rings and MP-rings,” J. Pure Appl. Algebra, 32, 11-20 (1984).
R. C. Courter, “Fully idempotent rings have regular centroids,” Proc. Amer. Math. Soc., 43, 293-296 (1974).
J. Dauns and K. H. Hofmann, “The representation of biregular rings by sheaves,” Math. Z., 91, 103-123 (1966).
J. Dauns and K. H. Hofmann, Representation of Rings by Sections. Mem. Amer. Math. Soc.Vol. 83, Providence, Rhode Island (1968).
G. De Marco and A. Orsatti, “Commutative rings in which every prime ideal is contained in a unique maximal ideal,” Proc. Amer. Math. Soc., 30, 459-466 (1971).
M. Desrochers and R. Raphael, “On finite extensions of hereditary regular rings,” Commun. Algebra, 21, No. 8, 2631-2636 (1993).
F. Dischinger, “Sur les anneaux fortement Π-réguliers,” C. R. Acad. Sci., 283, No. 8. A571-A573 (1976).
A. N. Dranishnikov, “The power substitution property for rings of complex and real functions on compact metric spaces,” Proc. Amer. Math. Soc., 123, No. 9, 2887-2893 (1995).
N. I. Dubrovin, “Lifting of idempotents in an algebraic algebra over a Henselian valuation ring,” Usp. Mat. Nauk, 41, No. 5, 173-174 (1986).
D. R. Farkas and R. L. Snider, “Locally finite-dimensional algebras,” Proc. Amer. Math. Soc., 81, No. 3, 369-372 (1981).
S. Feigelstock, “An embedding theorem for weakly regular rings and fully idempotent rings,” Comment. Math. Univ. St. Pauli, 27, No. 2, 101-103 (1978/79).
J. W. Fisher and R. L. Snider, “On the Von Neumann regularity of rings with regular prime factor rings,” Pacif. J. Math., 54, No. 1, 135-144 (1974).
J. W. Fisher and R. L. Snider, “Rings generated by their units,” J. Algebra, 42, 363-368 (1976).
J. A. Fraser and W. K. Nicholson, “Reduced p.p.-rings,” Math. Jpn., 34, No. 5, 715-725 (1989).
L. Fuchs and K. M. Rangaswamy, “On generalized regular rings,” Math. Z., 107, 71-81 (1968).
R. Gilmer, “Commutative rings with periodic multiplicative semigroup,” Commun. Algebra, 21, No. 11, 4025-4028 (1993).
R. Gilmer, “Zero-dimensional subrings of commutative rings,” In: Abelian Groups and Modules(A. Facchini and C. Menini, Eds.), Kluwer Acad. Publ., Dordrecht (1995), pp. 209-219.
K. R. Goodearl, Von Neumann Regular Rings, Pitman, London (1979).
K. R. Goodearl, “Surjective endomorphisms of finitely generated modules,” Commun. Algebra, 15, No. 3, 589-609 (1987).
K. R. Goodearl and D. E. Handelman, “Homogenization of regular rings of bounded index,” Pacif. J. Math., 84, No. 1, 63-78 (1979).
K. R. Goodearl and P. Menal, “Stable range one for rings with many units,” J. Pure Appl. Algebra, 54, 261-287 (1988).
K. R. Goodearl and J. Moncasi, “Cancellation of finitely generated modules over regular rings,” Osaka J. Math., 26, 679-685 (1989).
K. R. Goodearl and R. R. Warfield, “Algebras over zero-dimensional rings,” Math. Ann., 223, No. 2, 157-168 (1976).
V. Gupta, “Weakly Π-regular rings and group rings,” Math. J. Okayama Univ., 19, No. 2, 123-127 (1977).
V. Gupta, “A generalization of strongly regular rings,” Acta Math. Hung., 43, 57-61 (1984).
R. E. Hartwig, “On a theorem of Flanders,” Proc. Amer. Math. Soc., 85, No. 3. 613-616 (1982).
M. Henriksen, “Two classes of rings that are elementary divisor rings,” Arch. Math. (Basel), 31, No. 1, 182-193 (1973).
M. Henriksen, “Two classes of rings generated by their units,” J. Algebra, 31, No. 1, 182-193 (1974).
I. N. Herstein, Noncommutative Rings, John Wiley & Sons, New York (1968).
Y. Hirano, “Some studies on strongly Π-regular rings,” Math. J. Okayama Univ., 20, No. 2, 141-149 (1978).
Y. Hirano, “On fully right idempotent rings and direct sums of simple rings,” Math. J. Okayama Univ., 22, 43-49 (1980).
Y. Hirano, “On Π-regular rings with involution,” Math. J. Okayama Univ., 27, 45-47 (1985).
Y. Hirano, “Some characterizations of Π-regular rings of bounded index,” Math. J. Okayama Univ., 32, 97-101 (1990).
Y. Hirano and J. P. Park, “On self-injective strongly Π-regular rings,” Commun. Algebra 21, No. 1, 85-91 (1993).
C. Y. Hong, H. K. Kim and J. K. Park, “Rings with restricted descending chain conditions,” Commun. Algebra 25, No. 8, 2579-2584 (1997).
N. Jacobson, Structure of Rings, AMS, Providence (1968).
I. Kaplansky, “Topological representation of algebras,” Trans. Amer. Math. Soc., 68, 62-75 (1950).
K. Koh, “On one-sided ideals of a prime type,” Proc. Amer. Math. Soc., 28, No. 2, 321-329 (1971).
K. Koh, “On a representation of a strongly harmonic ring by sheaves,” Pacif. J. Math., 41, 459-468 (1972).
N. Lang, “On centroids of generalized regular rings,” Ann. Scu. Norm. Super. Pisa(3), 26, 573-585 (1972).
U. Leron, “Matrix methods in decompositions of modules,” Lin. Algebra Appl., 31, 159-171 (1980).
L. Lesieur, “Sur quelques classes d'anneaux non commutatifs,” Commun. Algebra 14, No. 8, 1481-1488 (1986).
J. Levitzki, “On the structure of algebraic algebras and related rings,” Trans. Amer. Math. Soc., 74, 384-409 (1953).
V. T. Markov, “On B-rings with polynomial identity,” Tr. Sem. Petrovskogo, No. 7, 232-238 (1981).
P. Menal, “On Π-regular rings whose primitive factor rings are Artinian,” J. Pure Appl. Algebra 20, No. 1, 71-78 (1981).
P. Menal, “Spectral Banach algebras of bounded index,” J. Algebra, 154, No. 1, 27-66 (1993).
K. Meyberg and B. Zimmermann-Huisgen, “Rings with descending chain condition on certain principal ideals,” Indag. Math., 39, No. 3, 225-229 (1977).
G. S. Monk, “A characterization of exchange rings,” Proc. Amer. Math. Soc. 35, No. 2, 349-353 (1972).
C. Nastasescu and N. Popescu, “Anneaux semi-artiniens,” Bull. Soc. Math. France, 96, 357-368 (1968).
E. Nauwelaerts and F. van Oystaeyen, “Zariski extensions and biregular rings,” Israel J. Math., 37, No. 4, 315-326 (1980).
W. K. Nicholson, “I-rings,” Trans. Amer. Math. Soc., 207, 361-373 (1975).
W. K. Nicholson, “Semiregular modules and rings,” Can. J. Math., 28, No. 5, 1105-1120 (1976).
W. K. Nicholson, “Lifting idempotents and exchange rings,” Trans. Amer. Math. Soc., 229, 269-278 (1977).
W. K. Nicholson, “On exchange rings,” Commun. Algebra, 25, No. 6, 1917-1918 (1997).
C. Nita, “Anneaux N-réguliers,” C. R. Acad. Sci., 268, A1241-A1243 (1969).
C. Nita, “Remarques sur les anneaux N-réguliers,” C. R. Acad. Sci., 271, A345-A348 (1970).
M. Ohori, “On strongly Π-regular rings and periodic rings,” Math. J. Okayama Univ., 19, No. 2, 123-127 (1977).
F. van Oystaeyen and J. van Geel, “Local-global results for regular rings,” Commun. Algebra 4, No. 9, 811-821 (1976).
E. Pardo, “Comparability, separativity, and exchange rings,” Commun. Algebra, 24, No. 9, 2915-2929 (1996).
E. Pardo, “Metric completions of ordered groups and K0 of exchange rings,” Trans. Amer. Math. Soc., 350, No. 3, 913-933 (1998).
R. S. Pierce, Modules over Commutative Regular Rings. Mem. Amer. Math. Soc.Vol. 70, Providence, Rhode Island (1967).
V. S. Ramamurthi, “Weakly regular rings,” Can. Math. Bull., 16, No. 3, 317-321 (1973).
K. M. Rangaswamy, “Endomorphism representations of Zorn rings and subclasses of rings with enough idempotents,” J. Reine Angew. Math., 276, 44-55 (1975).
R. Raphael, “Algebraic closures for some noncommutative rings,” Commun. Algebra, 17, No. 7, 1687-1708 (1989).
G. Shin, “Prime ideals and sheaf representation of a pseudosymmetric ring,” Trans. Amer. Math. Soc., 184, 43-60 (1973).
L. A. Skornyakov, Complemented Modular Lattices and Regular Rings, Oliver & Boyd, Edinburgh-London (1964).
K. I. Sonin, “Regular Laurent series rings,” Fund. Prikl. Mat., 1, No. 1, 315-317 (1995).
K. I. Sonin, “Regular skew Laurent series rings,” Fund. Prikl. Mat., 1, No. 2, 565-568 (1995).
K. I. Sonin, “Biregular Laurent series rings,” Vestn. Mosk. Univ., Ser. Mat., Mekh.No. 4, 20-22 (1997).
J. B. Srivastava and S. K. Shah, “Semilocal and semiregular group rings,” Nederl. Akad. Wetensch. Indag. Math., 42, No. 3, 347-352 (1980).
A. Steger, “Elementary factorization in Π-regular rings,” Can. J. Math., 18, 307-313 (1966).
S.-H. Sun, “On biregular rings and their duality,” J. Pure Appl. Algebra, 89, 329-337 (1993).
G. Szeto, “The localization of zero-dimensional rings,” Bull. Soc. Math. Belg., 28, 193-196 (1976).
G. Szeto, “On weakly biregular algebras,” Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 26, No. 9-10, 775-780 (1977).
K. Tanabe, “On rings whose Artinian modules are precisely Noetherian modules,” Commun. Algebra, 22, No. 10, 4023-4032 (1994).
K. Tanabe, “Some conditions for nil ideals and Π-regular ideals,” Commun. Algebra, 27, No. 6, 2573-2581 (1999).
D. V. Tjukavkin, “Rings all of whose one-sided ideals are generated by idempotents,” Commun. Algebra, 17, No. 5, 1193-1198 (1989).
D. V. Tjukavkin, “On regular rings of bounded index,” Mat. Zametki, 54, No. 1, 87-93 (1993).
H. Tominaga, “Some results on Π-regular rings of bounded index,” Math. J. Okayama Univ., 4, 135-141 (1955).
J. Trlifaj, “Steady rings may contain large sets of orthogonal idempotents,” In: Abelian Groups and Modules(A. Facchini and C. Menini, Eds.), Kluwer Acad. Publ., Dordrecht (1995), pp. 467-473.
H. Tsutsui, “Fully prime rings. II,” Commun. Algebra, 24, No. 9, 2981-2989 (1996).
A. A. Tuganbaev, “Distributive semiprime rings,” Mat. Zametki, 58, No. 5, 736-761 (1995).
A. A. Tuganbaev, “Bass rings and perfect rings,” Usp. Mat. Nauk, 51, No. 1, 173-174 (1996).
A. A. Tuganbaev, “Maximal submodules and locally perfect rings,” Mat. Zametki, 64, No. 1, 136-142 (1998).
A. A. Tuganbaev, Semidistributive Modules and Rings, Kluwer Academic Publishers, Dordrecht-Boston-London (1998).
A. A. Tuganbaev, Distributive Modules and Related Topics, Gordon and Breach, Amsterdam (1999).
J. Viola-Prioli, “On absolutely torsion-free rings,” Pacif. J. Math., 56, 275-283 (1975).
R. B. Warfield, “Exchange rings and decompositions of modules,” Math. Ann., 199, 31-36 (1972).
R. Wisbauer, “F-semiperfekte und perfekte Moduln in σ[M],” Math. Z., 173, 229-234 (1980).
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia (1991).
T. S. Wu, “Exchange rings with primitive factor rings Artinian,” Algebra Colloq., 3, No. 3, 225-230 (1996).
T. S. Wu and W. T. Tong, “Finitely generated projective modules over exchange rings,” Manuscr. Math., 86, No. 2, 149-157 (1995).
T. S. Wu and Y. Xu, “On the stable range condition of exchange rings,” Commun. Algebra, 25, No. 7, 2355-2363 (1997).
W. Xue, “Three questions on strongly Π-regular rings and regular rings,” Commun. Algebra 21, No. 2, 699-704 (1993).
W. Xue, “Semiregular modules and F-semiperfect modules,” Commun. Algebra, 23, No. 3, 1035-1046 (1995).
X. Yao, “Weakly right duo rings,” Pure Appl. Math. Sci., 21, 19-24 (1985).
H.-P. Yu, “On quasi-duo rings,” Glasgow Math. J., 37, 21-31 (1995).
H.-P. Yu, “On N0-quasi-continuous exchange rings,” Commun. Algebra, 23, No. 6, 2187-2197 (1995).
H.-P. Yu, “Stable range one for exchange rings,” J. Pure Appl. Algebra, 98, No. 1, 105-109 (1995).
H.-P. Yu, “On strongly Π-regular rings of stable range one,” Bull. Austral. Math. Soc., 51, No. 3, 433-437 (1995).
H.-P. Yu, “On the structure of exchange rings,” Commun. Algebra, 25, No. 2, 661-670 (1997).
J. Zelmanowitz, “Regular modules,” Trans. Amer. Math. Soc., 163, 341-355 (1972).
Rights and permissions
About this article
Cite this article
Tuganbaev, A.A. Semiregular, Weakly Regular, and Π-Regular Rings. Journal of Mathematical Sciences 109, 1509–1588 (2002). https://doi.org/10.1023/A:1013929008743
Issue Date:
DOI: https://doi.org/10.1023/A:1013929008743