( 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

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.