On commutative rings whose prime ideals are direct sums of cyclics

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 -