( n , d ) -injective covers, n -coherent rings, and ( n , d ) -rings

Weiqing Li; Baiyu Ouyang

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 2, page 289-304
  • ISSN: 0011-4642

Abstract

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It is known that a ring R is left Noetherian if and only if every left R -module has an injective (pre)cover. We show that ( 1 ) if R is a right n -coherent ring, then every right R -module has an ( n , d ) -injective (pre)cover; ( 2 ) if R is a ring such that every ( n , 0 ) -injective right R -module is n -pure extending, and if every right R -module has an ( n , 0 ) -injective cover, then R is right n -coherent. As applications of these results, we give some characterizations of ( n , d ) -rings, von Neumann regular rings and semisimple rings.

How to cite

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Li, Weiqing, and Ouyang, Baiyu. "$(n,d)$-injective covers, $n$-coherent rings, and $(n,d)$-rings." Czechoslovak Mathematical Journal 64.2 (2014): 289-304. <http://eudml.org/doc/261991>.

@article{Li2014,
abstract = {It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $(1)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $(n,d)$-injective (pre)cover; $(2)$ if $R$ is a ring such that every $(n,0)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $(n,0)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $(n,d)$-rings, von Neumann regular rings and semisimple rings.},
author = {Li, Weiqing, Ouyang, Baiyu},
journal = {Czechoslovak Mathematical Journal},
keywords = {cover; envelope; $n$-coherent ring; $(n,d)$-injective; $(n,d)$-ring; injective covers; injective precovers; injective envelopes; coherent rings; injective modules; -rings},
language = {eng},
number = {2},
pages = {289-304},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$(n,d)$-injective covers, $n$-coherent rings, and $(n,d)$-rings},
url = {http://eudml.org/doc/261991},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Li, Weiqing
AU - Ouyang, Baiyu
TI - $(n,d)$-injective covers, $n$-coherent rings, and $(n,d)$-rings
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 289
EP - 304
AB - It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $(1)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $(n,d)$-injective (pre)cover; $(2)$ if $R$ is a ring such that every $(n,0)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $(n,0)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $(n,d)$-rings, von Neumann regular rings and semisimple rings.
LA - eng
KW - cover; envelope; $n$-coherent ring; $(n,d)$-injective; $(n,d)$-ring; injective covers; injective precovers; injective envelopes; coherent rings; injective modules; -rings
UR - http://eudml.org/doc/261991
ER -

References

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  1. Anderson, F. W., Fuller, K. R., Rings and Categories of Modules, (2nd ed.). Graduate Texts in Mathematics 13 Springer, New York (1992). (1992) Zbl0765.16001MR1245487
  2. Bican, L., Bashir, R. El, Enochs, E., 10.1017/S0024609301008104, Bull. Lond. Math. Soc. 33 (2001), 385-390. (2001) Zbl1029.16002MR1832549DOI10.1017/S0024609301008104
  3. Chen, J., Ding, N., 10.1080/00927879608825742, Commun. Algebra 24 (1996), 3211-3216. (1996) Zbl0877.16010MR1402554DOI10.1080/00927879608825742
  4. Costa, D. L., 10.1080/00927879408825061, Commun. Algebra 22 (1994), 3997-4011. (1994) Zbl0814.13010MR1280104DOI10.1080/00927879408825061
  5. Damiano, R. F., 10.2140/pjm.1979.81.349, Pac. J. Math. 81 (1979), 349-369. (1979) Zbl0415.16021MR0547604DOI10.2140/pjm.1979.81.349
  6. Ding, N., 10.1080/00927879608825646, Commun. Algebra 24 (1996), 1459-1470. (1996) Zbl0863.16005MR1380605DOI10.1080/00927879608825646
  7. Enochs, E. E., 10.1007/BF02760849, Isr. J. Math. 39 (1981), 189-209. (1981) Zbl0464.16019MR0636889DOI10.1007/BF02760849
  8. Enochs, E. E., Jenda, O. M. G., Relative Homological Algebra, De Gruyter Expositions in Mathematics 30 Walter de Gruyter, Berlin (2000). (2000) Zbl0952.13001MR1753146
  9. Facchini, A., Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Mathematics 167 Birkhäuser, Basel (1998). (1998) Zbl0930.16001MR1634015
  10. Mao, L., Ding, N., 10.1080/00927870600649111, Commun. Algebra 34 (2006), 2403-2418. (2006) Zbl1104.16002MR2240382DOI10.1080/00927870600649111
  11. Pinzon, K., 10.1080/00927870801952694, Commun. Algebra 36 (2008), 2186-2194. (2008) Zbl1162.16003MR2418384DOI10.1080/00927870801952694
  12. Rotman, J. J., An Introduction to Homological Algebra, (2nd ed.). Universitext Springer, New York (2009). (2009) Zbl1157.18001MR2455920
  13. Stenström, B., 10.1112/jlms/s2-2.2.323, J. Lond. Math. Soc., II. Ser. 2 (1970), 323-329. (1970) Zbl0194.06602MR0258888DOI10.1112/jlms/s2-2.2.323
  14. Xu, J., 10.1007/BFb0094173, Lecture Notes in Mathematics 1634 Springer, Berlin (1996). (1996) Zbl0860.16002MR1438789DOI10.1007/BFb0094173
  15. Ming, R. Yue Chi, 10.1007/BF01301145, Monatsh. Math. 95 (1983), 25-32. (1983) MR0697346DOI10.1007/BF01301145
  16. Zhou, D., 10.1081/AGB-120037230, Commun. Algebra 32 (2004), 2425-2441. (2004) Zbl1089.16001MR2100480DOI10.1081/AGB-120037230
  17. Zhou, D. X., 10.1007/s10114-009-6385-7, Acta Math. Sin., Engl. Ser. 25 (2009), 1567-1582. (2009) Zbl1215.16016MR2544301DOI10.1007/s10114-009-6385-7

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