Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
strongly regular; $p$-injective; $\operatorname{YJ}$-injective; biregular; von Neumann regular
strongly regular; $p$-injective; $\operatorname{YJ}$-injective; biregular; von Neumann regular
Summary:
The following results are proved for a ring $A$: (1) If $A$ is a fully right idempotent ring having a classical left quotient ring $Q$ which is right quasi-duo, then $Q$ is a strongly regular ring; (2) $A$ has a classical left quotient ring $Q$ which is a finite direct sum of division rings iff $A$ is a left $\operatorname{TC}$-ring having a reduced maximal right ideal and satisfying the maximum condition on left annihilators; (3) Let $A$ have the following properties: (a) each maximal left ideal of $A$ is either a two-sided ideal of $A$ or an injective left $A$-module; (b) for every maximal left ideal $M$ of $A$ which is a two-sided ideal, $A/M_A$ is flat. Then, $A$ is either strongly regular or left self-injective regular with non-zero socle; (4) $A$ is strongly regular iff $A$ is a semi-prime left or right quasi-duo ring such that for every essential left ideal $L$ of $A$ which is a two-sided ideal, $A/L_A$ is flat; (5) $A$ prime ring containing a reduced minimal left ideal must be a division ring; (6) A commutative ring is quasi-Frobenius iff it is a $\operatorname{YJ}$-injective ring with maximum condition on annihilators.
References:
[1] Arens R.F., Kaplansky I.: Topological representations of algebras. Trans. Amer. Math. Soc. 63 (1948), 457-481. MR 0025453
[2] Armendariz E.P., Fisher J.W.: Regular $P.I.$-rings. Proc. Amer. Math. Soc. 39 (1973), 247-251. MR 0313305 | Zbl 0264.16010
[3] Baccella G.: Generalized $V$-rings and von Neumann regular rings. Rend. Sem. Mat. Univ. Padova 72 (1984), 117-133. MR 0778337 | Zbl 0547.16006
[4] Beidar K., Wisbauer R.: Property semi-prime self-pp-modules. Comm. Algebra 23 (1995), 841-861. MR 1316735
[5] Chase S.U.: Direct product of modules. Trans. Amer. Math. Soc. 97 (1960), 457-473. MR 0120260
[6] Chen J.L., Ding N.Q.: On general principally injective rings. Comm. Algebra 27 (1999), 2097-2116. MR 1683854 | Zbl 0923.16001
[7] Chen J.L., Zhou Y.Q., Zhu Z.M.: GP-injective rings need not be $p$-injective. Comm. Algebra 33 (2005), 2395-2402. MR 2153231 | Zbl 1076.16003
[8] Faith C.: Algebra II: Ring Theory. Grundlehren der Mathematischen Wissenschaft, no. 191, Springer, Berlin-New York, 1976. MR 0427349
[9] Faith C.: Ring and things and a fine array of twentieth century associative algebra. Mathematical Surveys and Monographs, 65, American Mathematical Society, Providence, 1999. MR 1657671
[10] Goodearl K.: Ring Theory: Nonsingular Rings and Modules. Pure and Applied Mathematics, no. 33, Marcel Dekker, New York, 1976. MR 0429962 | Zbl 0336.16001
[11] Goodearl K.: Von Neumann Regular Rings. Pitman, Boston, 1979. MR 0533669 | Zbl 0841.16008
[12] Hirano Y.: On non-singular $p$-injective rings. Publ. Math. 38 (1994), 455-461.
[13] Kasch F.: Modules and Rings. London Mathematical Society Monographs, 17, Academic Press, London-New York, 1982. MR 0667346 | Zbl 0832.16002
[14] Kim N.K., Nam S.B., Kim J.Y.: On simple singular GP-injective modules. Comm. Algebra 27 (1999), 2087-2096. MR 1683853 | Zbl 0923.16008
[15] Matlis E.: Injective modules over Noetherian rings. Pacific J. Math. 8 (1958), 511-528. MR 0099360 | Zbl 0084.26601
[16] Nicholson W.K, Yousif M.F.: Principally injective rings. J. Algebra 174 (1995), 77-93. MR 1332860 | Zbl 0839.16004
[17] Puninski G., Wisbauer R., Yousif M.F.: On $p$-injective rings. Glasgow Math. J. 37 (1995), 373-378. MR 1355393 | Zbl 0847.16005
[18] Sandomierski F.: Semi-simple maximal quotient rings. Trans. Amer. Math. Soc. 128 (1967), 112-120. MR 0214624
[19] Storrer H.H.: A note on quasi-Frobenius rings and ring epimorphism. Canad. Math. Bull. 12 (1969), 287-292. MR 0251075
[20] Tuganbaev A.: Rings close to regular. Mathematics and its Applications, 545, Kluwer Academic Publishers, Dordrecht, 2002. MR 1958361 | Zbl 1120.16012
[21] Wisbauer R.: Foundations of module and ring theory. Gordon and Breach, New York, 1991. MR 1144522 | Zbl 0746.16001
[22] Xue W.M.: A note on YJ-injectivity. Riv. Mat. Univ. Parma (6) 1 (1998), 31-37. MR 1680954 | Zbl 0929.16002
[23] Xue W.M.: Rings related to quasi-Frobenius rings. Algebra Colloq. 5 (1998), 471-480. MR 1683093 | Zbl 0937.16028
[24] Yousif M.F.: On SI-modules. Math. J. Okayama Univ. 28 (1986), 133-146. MR 0885022
[25] Yu H.P.: On quasi-duo rings. Glasgow Math. J. 37 (1995), 21-31. MR 1316960 | Zbl 0819.16001
[26] Yue Chi Ming R.: A note on singular ideals. Tôhoku Math. J. 21 (1969), 337-342. MR 0252444 | Zbl 0164.34702
[27] Yue Chi Ming R.: On annihilator ideals. Math. J. Okayama Univ. 19 (1976), 51-53. MR 0435130 | Zbl 0348.16008
[28] Yue Chi Ming R.: On generalizations of $V$-rings and regular rings. Math. J. Okayama Univ. 20 (1978), 123-129. MR 0519559 | Zbl 0402.16014
[29] Yue Chi Ming R.: A remark on decomposable modules. Publ. Inst. Math. (Beograd) 25 (39) (1979), 101-104. MR 0542830 | Zbl 0408.16018
[30] Yue Chi Ming R.: On $V$-rings and prime rings. J. Algebra 62 (1980), 13-20. MR 0561114 | Zbl 0429.16018
[31] Yue Chi Ming R.: On von Neumann regular rings, VI. Rend. Sem. Mat. Univ. Politec. Torino 39 (1981), 75-84. MR 0706046 | Zbl 0507.16012
[32] Yue Chi Ming R.: On regular rings and Artinian rings. Riv. Mat. Univ. Parma (4) 8 (1982), 443-452. MR 0706868 | Zbl 0516.16006
[33] Yue Chi Ming R.: On quasi-injectivity and von Neumann regularity. Monatsh. Math. 95 (1983), 25-32. MR 0697346 | Zbl 0498.16008
[34] Yue Chi Ming R.: On regular rings and self-injective rings, II. Glas. Mat. 18 (38) (1983), 221-229. MR 0733161 | Zbl 0528.16006
[35] Yue Chi Ming R.: On von Neumann regular rings, X. Collect. Math. 34 (1983), 81-94. MR 0747858 | Zbl 0544.16006
[36] Yue Chi Ming R.: Annihilators and strongly regular rings. Rend. Sem. Fac. Sci. Cagliari 57 (1987), 51-59. MR 0976539 | Zbl 0683.16010
[37] Yue Chi Ming R.: On $p$-injectivity, YJ-injectivity and quasi-Frobeniusean rings. Comment. Math. Univ. Carolin. 43 (2002), 33-42. MR 1903305 | Zbl 1068.16004
[38] Zhang J.L., Wu J.: Generalizations of principal injectivity. Algebra Colloq. 6 (1999), 277-282. MR 1809647 | Zbl 0949.16002
Search
Partner of
