On commutative rings whose prime ideals are direct sums of cyclics

M. Behboodi; A. Moradzadeh-Dehkordi

Archivum Mathematicum (2012)

  • Volume: 048, Issue: 4, page 291-299
  • ISSN: 0044-8753

Abstract

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In this paper we study commutative rings R whose prime ideals are direct sums of cyclic modules. In the case R is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring ( R , ) , the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R -modules; (2) = λ Λ R w λ where Λ is an index set and R / Ann ( w λ ) is a principal ideal ring for each λ Λ ; (3) Every prime ideal of R is a direct sum of at most | Λ | cyclic R -modules where Λ is an index set and = λ Λ R w λ ; and (4) Every prime ideal of R is a summand of a direct sum of cyclic R -modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring ( R , ) is a direct sum of (at most n ) principal ideals, it suffices to test only the maximal ideal .

How to cite

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Behboodi, M., and Moradzadeh-Dehkordi, A.. "On commutative rings whose prime ideals are direct sums of cyclics." Archivum Mathematicum 048.4 (2012): 291-299. <http://eudml.org/doc/251357>.

@article{Behboodi2012,
abstract = {In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R, \mathcal \{M\})$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) $\{\mathcal \{M\}\}=\bigoplus _\{\lambda \in \Lambda \}Rw_\{\lambda \}$ where $\Lambda $ is an index set and $R/\{\operatorname\{Ann\}\}(w_\{\lambda \})$ is a principal ideal ring for each $\lambda \in \Lambda $; (3) Every prime ideal of $R$ is a direct sum of at most $|\Lambda |$ cyclic $R$-modules where $\Lambda $ is an index set and $\{\mathcal \{M\}\}=\bigoplus _\{\lambda \in \Lambda \}Rw_\{\lambda \}$; and (4) Every prime ideal of $R$ is a summand of a direct sum of cyclic $R$-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring $(R, \mathcal \{M\})$ is a direct sum of (at most $n$) principal ideals, it suffices to test only the maximal ideal $\mathcal \{M\}$.},
author = {Behboodi, M., Moradzadeh-Dehkordi, A.},
journal = {Archivum Mathematicum},
keywords = {prime ideals; cyclic modules; local rings; principal ideal rings; prime ideals; cyclic modules; local rings; principal ideal rings},
language = {eng},
number = {4},
pages = {291-299},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On commutative rings whose prime ideals are direct sums of cyclics},
url = {http://eudml.org/doc/251357},
volume = {048},
year = {2012},
}

TY - JOUR
AU - Behboodi, M.
AU - Moradzadeh-Dehkordi, A.
TI - On commutative rings whose prime ideals are direct sums of cyclics
JO - Archivum Mathematicum
PY - 2012
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 048
IS - 4
SP - 291
EP - 299
AB - In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R, \mathcal {M})$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) ${\mathcal {M}}=\bigoplus _{\lambda \in \Lambda }Rw_{\lambda }$ where $\Lambda $ is an index set and $R/{\operatorname{Ann}}(w_{\lambda })$ is a principal ideal ring for each $\lambda \in \Lambda $; (3) Every prime ideal of $R$ is a direct sum of at most $|\Lambda |$ cyclic $R$-modules where $\Lambda $ is an index set and ${\mathcal {M}}=\bigoplus _{\lambda \in \Lambda }Rw_{\lambda }$; and (4) Every prime ideal of $R$ is a summand of a direct sum of cyclic $R$-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring $(R, \mathcal {M})$ is a direct sum of (at most $n$) principal ideals, it suffices to test only the maximal ideal $\mathcal {M}$.
LA - eng
KW - prime ideals; cyclic modules; local rings; principal ideal rings; prime ideals; cyclic modules; local rings; principal ideal rings
UR - http://eudml.org/doc/251357
ER -

References

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  2. Behboodi, M., Ghorbani, A., Moradzadeh–Dehkordi, A., 10.1016/j.jalgebra.2011.08.017, J. Algebra 345 (2011), 257–265. (2011) Zbl1244.13008MR2842065DOI10.1016/j.jalgebra.2011.08.017
  3. Behboodi, M., Ghorbani, A., Moradzadeh–Dehkordi, A., Shojaee, S. H., On left Köthe rings and a generalization of the Köthe–Cohen–Kaplansky theorem, Proc. Amer. Math. Soc. (to appear). MR2530766
  4. Behboodi, M., Shojaee, S. H., Commutative local rings whose ideals are direct sums of cyclic modules, submitted. 
  5. Cohen, I. S., 10.1215/S0012-7094-50-01704-2, Duke Math. J. 17 (1950), 27–42. (1950) Zbl0041.36408MR0033276DOI10.1215/S0012-7094-50-01704-2
  6. Cohen, I. S., Kaplansky, I., 10.1007/BF01179851, Math. Z. 54 (1951), 97–101. (1951) Zbl0043.26702MR0043073DOI10.1007/BF01179851
  7. Kaplansky, I., 10.1090/S0002-9947-1949-0031470-3, Trans. Amer. Math. Soc. 66 (1949), 464–491. (1949) Zbl0036.01903MR0031470DOI10.1090/S0002-9947-1949-0031470-3
  8. Köthe, G., 10.1007/BF01201343, Math. Z. 39 (1935), 31–44. (1935) MR1545487DOI10.1007/BF01201343
  9. Warfield, R. B. Jr.,, 10.1090/S0002-9939-1969-0242886-2, Proc. Amer. Math. Soc. 22 (1969), 460–465. (1969) Zbl0176.31401MR0242886DOI10.1090/S0002-9939-1969-0242886-2

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