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Published: 10 December 2022
Last updated: 28 January 2023

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1

Tuganbaev, A. A. "Bass rings and perfect rings." Russian Mathematical Surveys 51, no. 1 (February 28, 1996): 173–74. http://dx.doi.org/10.1070/rm1996v051n01abeh002767.

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2

Martinez, Jorge. "Categorically perfect rings." Communications in Algebra 23, no. 10 (January 1995): 3641–51. http://dx.doi.org/10.1080/00927879508825423.

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3

Năstăsescu, C., L. Dăuş, and B. Torrecillas. "Graded Semiartinian Rings: Graded Perfect Rings." Communications in Algebra 31, no. 9 (January 9, 2003): 4525–31. http://dx.doi.org/10.1081/agb-120022807.

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4

Udar, Dinesh, R. K. Sharma, and J. B. Srivastava. "Restricted Perfect Group Rings." Communications in Algebra 44, no. 9 (May 19, 2016): 4097–103. http://dx.doi.org/10.1080/00927872.2015.1087018.

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5

Fuchs, László, and Luigi Salce. "Almost perfect commutative rings." Journal of Pure and Applied Algebra 222, no. 12 (December 2018): 4223–38. http://dx.doi.org/10.1016/j.jpaa.2018.02.029.

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6

Ánh, P. N., N. V. Loi, and D. V. Thanh. "Perfect rings without identity." Communications in Algebra 19, no. 4 (January 1991): 1069–82. http://dx.doi.org/10.1080/00927879108824190.

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7

Beattie, Margaret, and Eric Jespers. "On perfect graded rings." Communications in Algebra 19, no. 8 (January 1991): 2363–71. http://dx.doi.org/10.1080/00927879108824264.

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8

Camillo, Victor P., and Weimin Xue. "On quasi-perfect rings." Communications in Algebra 19, no. 10 (January 1991): 2841–50. http://dx.doi.org/10.1080/00927879108824297.

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9

Amini, A., B. Amini, M. Ershad, and H. Sharif. "On Generalized Perfect Rings." Communications in Algebra 35, no. 3 (February 27, 2007): 953–63. http://dx.doi.org/10.1080/00927870601115880.

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10

Jhilal, Abdellatif, and Najib Mahdou. "On Strongn-Perfect Rings." Communications in Algebra 38, no. 3 (March 16, 2010): 1057–65. http://dx.doi.org/10.1080/00927870902828769.

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11

Gupta, Ram Niwas. "Characterization of rings whose classical quotient rings are perfect rings." Publicationes Mathematicae Debrecen 17, no. 1-4 (July 1, 2022): 215–22. http://dx.doi.org/10.5486/pmd.1970.17.1-4.25.

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12

Herbera, Dolors, and Ahmad Shamsuddin. "On Self-Injective Perfect Rings." Canadian Mathematical Bulletin 39, no. 1 (March 1, 1996): 55–58. http://dx.doi.org/10.4153/cmb-1996-007-8.

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AbstractLet R be a left and right perfect right self-injective ring. It is shown that if the radical of R is countably generated as a left ideal then R is quasi-Frobenius. It is also shown that the same conclusion can be drawn if r(A ∩ B) = r(A) + r(B) for all left ideals A and B of R.
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13

VAŠ, LIA. "PERFECT SYMMETRIC RINGS OF QUOTIENTS." Journal of Algebra and Its Applications 08, no. 05 (October 2009): 689–711. http://dx.doi.org/10.1142/s021949880900359x.

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Perfect Gabriel filters of right ideals and their corresponding right rings of quotients have the desirable feature that every module of quotients is determined solely by the right ring of quotients. On the other hand, symmetric rings of quotients have a symmetry that mimics the commutative case. In this paper, we study rings of quotients that combine these two desirable properties. We define the symmetric versions of a right perfect ring of quotients and a right perfect Gabriel filter — the perfect symmetric ring of quotients and the perfect symmetric Gabriel filter and study their properties. Then we prove that the standard construction of the total right ring of quotients [Formula: see text] can be adapted to the construction of the largest perfect symmetric ring of quotients — the total symmetric ring of quotients [Formula: see text]. We also demonstrate that Morita's construction of [Formula: see text] can be adapted to the construction of [Formula: see text].
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14

Ndiaye, Mohameth. "On Perfect Commutative EIFA-rings." British Journal of Mathematics & Computer Science 4, no. 8 (January 10, 2014): 1166–69. http://dx.doi.org/10.9734/bjmcs/2014/7863.

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15

Bazzoni, Silvana, and Leonid Positselski. "S-almost perfect commutative rings." Journal of Algebra 532 (August 2019): 323–56. http://dx.doi.org/10.1016/j.jalgebra.2019.05.018.

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16

Nashier, Budh, and Warren Nichols. "A note on perfect rings." Manuscripta Mathematica 70, no. 1 (December 1991): 307–10. http://dx.doi.org/10.1007/bf02568380.

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17

Nicholson, W. K., and M. F. Yousif. "On perfect simple-injective rings." Proceedings of the American Mathematical Society 125, no. 4 (1997): 979–85. http://dx.doi.org/10.1090/s0002-9939-97-03678-2.

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18

Abrams, Gene, and Frank Anderson. "Perfect Rings: A Local Approach." Communications in Algebra 33, no. 11 (October 2005): 4171–75. http://dx.doi.org/10.1080/00927870500261488.

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19

Amini, Babak, Afshin Amini, and Majid Ershad. "Almost-Perfect Rings and Modules." Communications in Algebra 37, no. 12 (November 24, 2009): 4227–40. http://dx.doi.org/10.1080/00927870902828918.

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20

Escalona, Miguel Angel Fortes, Inmaculada De Las Peñas Cabrera, and Esperanza Sánchez Campos. "ON PERFECT ASSOCIATIVE PAIRS. APPLICATION TO NONUNITAL PERFECT RINGS." Communications in Algebra 29, no. 2 (January 31, 2001): 625–38. http://dx.doi.org/10.1081/agb-100001529.

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21

Bunina, E. I., and A. S. Dobrokhotova-Maykova. "Elementary equivalence of incidence rings over semi-perfect rings." Journal of Mathematical Sciences 185, no. 2 (July 26, 2012): 199–206. http://dx.doi.org/10.1007/s10958-012-0909-x.

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22

Positselski, Leonid. "Contramodules over pro-perfect topological rings." Forum Mathematicum 34, no. 1 (November 30, 2021): 1–39. http://dx.doi.org/10.1515/forum-2021-0010.

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Abstract For four wide classes of topological rings R \mathfrak{R} , we show that all flat left R \mathfrak{R} -contramodules have projective covers if and only if all flat left R \mathfrak{R} -contramodules are projective if and only if all left R \mathfrak{R} -contramodules have projective covers if and only if all descending chains of cyclic discrete right R \mathfrak{R} -modules terminate if and only if all the discrete quotient rings of R \mathfrak{R} are left perfect. Three classes of topological rings for which this holds are the complete, separated topological associative rings with a base of neighborhoods of zero formed by open two-sided ideals such that either the ring is commutative, or it has a countable base of neighborhoods of zero, or it has only a finite number of semisimple discrete quotient rings. The fourth class consists of all the topological rings with a base of neighborhoods of zero formed by open right ideals which have a closed two-sided ideal with certain properties such that the quotient ring is a topological product of rings from the previous three classes. The key technique on which the proofs are based is the contramodule Nakayama lemma for topologically T-nilpotent ideals.
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23

Zhou, Yiqiang. "A Characterization of Left Perfect Rings." Canadian Mathematical Bulletin 38, no. 3 (September 1, 1995): 382–84. http://dx.doi.org/10.4153/cmb-1995-055-6.

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AbstractIn this note, we show that a ring R is a left perfect ring if and only if every generating set of each left R-module contains a minimal generating set. This result gives a positive answer to a question on left perfect rings raised by Nashier and Nichols.
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24

Xue, W. "Characterizations of semiperfect and perfect rings." Publicacions Matemàtiques 40 (January 1, 1996): 115–25. http://dx.doi.org/10.5565/publmat_40196_08.

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25

Ding, Nanqing, and Jianlong Chen. "On a characterization of perfect rings*." Communications in Algebra 27, no. 2 (January 1999): 785–91. http://dx.doi.org/10.1080/00927879908826461.

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26

Bhatt, Bhargav, Srikanth B. Iyengar, and Linquan Ma. "Regular rings and perfect(oid) algebras." Communications in Algebra 47, no. 6 (April 17, 2019): 2367–83. http://dx.doi.org/10.1080/00927872.2018.1524009.

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27

Enochs, Edgar E., Overtoun M. G. Jenda, and J. A. López-Ramos. "DUALIZING MODULES AND n-PERFECT RINGS." Proceedings of the Edinburgh Mathematical Society 48, no. 1 (February 2005): 75–90. http://dx.doi.org/10.1017/s0013091503001056.

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AbstractIn this article we extend the results about Gorenstein modules and Foxby duality to a non-commutative setting. This is done in §3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote §2 of the paper to a consideration of such rings. Finally, in the paper’s final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of §3, i.e. when we have a dualizing module.AMS 2000 Mathematics subject classification: Primary 16D20
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28

Kedlaya, Kiran S. "Automorphisms of perfect power series rings." Journal of Algebra 511 (October 2018): 358–63. http://dx.doi.org/10.1016/j.jalgebra.2018.05.035.

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29

Burkholder, Douglas G. "Azumaya rings with locally perfect centers." Journal of Algebra 103, no. 2 (October 1986): 606–18. http://dx.doi.org/10.1016/0021-8693(86)90155-9.

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30

Faticoni, Theodore G. "Semi-perfect FPF rings and applications." Journal of Algebra 107, no. 2 (May 1987): 297–315. http://dx.doi.org/10.1016/0021-8693(87)90092-5.

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31

Lotfi, Shokufeh, Mohammad Reza R. Moghaddam, and S. Mostafa Taheri. "SOME PROPERTIES OF PERFECT LIE RINGS." JP Journal of Algebra, Number Theory and Applications 39, no. 2 (April 7, 2017): 247–60. http://dx.doi.org/10.17654/nt039020247.

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32

Kuratomi, Yosuke, and Chaehoon Chang. "Lifting Modules Over Right Perfect Rings." Communications in Algebra 35, no. 10 (September 21, 2007): 3103–9. http://dx.doi.org/10.1080/00927870701405132.

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33

Rump, W. "I-radicals and right perfect rings." Ukrainian Mathematical Journal 59, no. 7 (July 2007): 1114–19. http://dx.doi.org/10.1007/s11253-007-0072-6.

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34

Tuganbaev, A. A. "Maximal submodules and locally perfect rings." Mathematical Notes 64, no. 1 (July 1998): 116–20. http://dx.doi.org/10.1007/bf02307202.

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35

MAIMANI, H. R., M. R. POURNAKI, and S. YASSEMI. "WEAKLY PERFECT GRAPHS ARISING FROM RINGS." Glasgow Mathematical Journal 52, no. 3 (March 22, 2010): 417–25. http://dx.doi.org/10.1017/s0017089510000108.

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AbstractA graph is called weakly perfect if its chromatic number equals its clique number. In this paper a new class of weakly perfect graphs arising from rings are presented and an explicit formula for the chromatic number of such graphs is given.
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36

Ercolanoni, Sofia, and Alberto Facchini. "Projective covers over local rings." Annali di Matematica Pura ed Applicata (1923 -) 200, no. 6 (March 19, 2021): 2631–44. http://dx.doi.org/10.1007/s10231-021-01095-5.

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AbstractWe describe the structure of the projective cover of a module $$M_R$$ M R over a local ring R and its relation with minimal sets of generators of $$M_R$$ M R . The behaviour of local right perfect rings is completely different from the behaviour of local rings that are not right perfect.
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37

BENNIS, DRISS, and NAJIB MAHDOU. "ON n-PERFECT RINGS AND COTORSION DIMENSION." Journal of Algebra and Its Applications 08, no. 02 (April 2009): 181–90. http://dx.doi.org/10.1142/s0219498809003266.

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A ring is called n-perfect (n ≥ 0), if every flat module has projective dimension less or equal than n. In this paper, we show that the n-perfectness relates, via homological approach, some homological dimensions of rings. We study n-perfectness in some known ring construction. Finally, several examples of n-perfect rings satisfying special conditions are given.
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38

Trlifaj, Jan. "The Dual Baer Criterion for non-perfect rings." Forum Mathematicum 32, no. 3 (May 1, 2020): 663–72. http://dx.doi.org/10.1515/forum-2019-0028.

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AbstractBaer’s Criterion for Injectivity is a useful tool of the theory of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for the non-right perfect ones is a complex problem (first formulated by C. Faith [Algebra. II. Ring Theory, Springer, Berlin, 1976]). Recently, it has turned out that there are two classes of non-right perfect rings: (1) those for which DBC fails in ZFC, and (2) those for which DBC is independent of ZFC. First examples of rings in the latter class were constructed in [J. Trlifaj, Faith’s problem on R-projectivity is undecidable, Proc. Amer. Math. Soc. 147 2019, 2, 497–504]; here, we show that this class contains all small semiartinian von Neumann regular rings with primitive factors artinian.
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39

Facchini, Alberto, and Catia Parolin. "Rings whose proper factors are right perfect." Colloquium Mathematicum 122, no. 2 (2011): 191–202. http://dx.doi.org/10.4064/cm122-2-4.

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40

Mirghadim, S. M. Saadat, R. Nikandish, and M. J. Nikmehr. "Perfect unit graphs of commutative Artinian rings." Afrika Matematika 32, no. 5-6 (January 18, 2021): 891–96. http://dx.doi.org/10.1007/s13370-020-00868-0.

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41

Trlifaj, Jan. "Strong Incompactness for Some Non-Perfect Rings." Proceedings of the American Mathematical Society 123, no. 1 (January 1995): 21. http://dx.doi.org/10.2307/2160604.

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42

Positselski, Leonid, and Jan Šťovíček. "Topologically semisimple and topologically perfect topological rings." Publicacions Matemàtiques 66 (July 1, 2022): 457–540. http://dx.doi.org/10.5565/publmat6622202.

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43

Chang, Chae-Hoon. "X-LIFTING MODULES OVER RIGHT PERFECT RINGS." Bulletin of the Korean Mathematical Society 45, no. 1 (February 29, 2008): 59–66. http://dx.doi.org/10.4134/bkms.2008.45.1.059.

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44

Ouarghi, Khalid. "Weakly injective dimension and Almost perfect rings." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 5 (May 25, 2016): 6185–90. http://dx.doi.org/10.24297/jam.v12i5.211.

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In this paper, we study the weak-injective dimension and we characterize the global weak-injective dimension of rings. After we study the transfer of the global weak-injective dimension in some known ring construction. Finely westudy the transfer of almost perfect property in pullback and D+M constructions.
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45

Shin, Jong Moon, and Chae-Hoon Chang. "X-LIFTING MODULES OVER RIGHT PERFECT RINGS." Pure and Applied Mathematics 21, no. 2 (May 31, 2014): 95–103. http://dx.doi.org/10.7468/jksmeb.2014.21.2.95.

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46

Rincón-Mejía, H. A. "The lattice $R$-tors for perfect rings." Publicacions Matemàtiques 33 (January 1, 1989): 17–35. http://dx.doi.org/10.5565/publmat_33189_02.

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47

Zhou, Yiqiang. "Erratum: A Characterization of Left Perfect Rings." Canadian Mathematical Bulletin 45, no. 3 (September 1, 2002): 448. http://dx.doi.org/10.4153/cmb-2002-047-7.

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48

Abyzov, A. N., T. C. Quynh, and D. D. Tai. "Dual automorphism-invariant modules over perfect rings." Siberian Mathematical Journal 58, no. 5 (September 2017): 743–51. http://dx.doi.org/10.1134/s0037446617050019.

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49

Ebrahimi, Sh, A. Tehranian, and R. Nikandish. "On perfect annihilator graphs of commutative rings." Discrete Mathematics, Algorithms and Applications 12, no. 05 (August 27, 2020): 2050060. http://dx.doi.org/10.1142/s1793830920500603.

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Let [Formula: see text] be a commutative ring with identity, and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The annihilator graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study the perfectness of annihilator graphs of a vast range of rings. Indeed, it is shown that if [Formula: see text] is reduced with finitely many minimal primes or nonreduced, then [Formula: see text] is perfect.
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50

Xue, Weimin. "A note on perfect self-injective rings(*)." Communications in Algebra 24, no. 2 (January 1996): 749–55. http://dx.doi.org/10.1080/00927879608825597.

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