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Keywords:
prime ideals; cyclic modules; local rings; principal ideal rings
prime ideals; cyclic modules; local rings; principal ideal rings
Summary:
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 $\mathcal{M}$.
References:
[1] Anderson, D. D.: A note on minimal prime ideals. Proc. Amer. Math. Soc. 122 (1994), 13–14. DOI 10.1090/S0002-9939-1994-1191864-2 | MR 1191864 | Zbl 0841.13001
[2] Behboodi, M., Ghorbani, A., Moradzadeh–Dehkordi, A.: Commutative Noetherian local rings whose ideals are direct sums of cyclic modules. J. Algebra 345 (2011), 257–265. DOI 10.1016/j.jalgebra.2011.08.017 | MR 2842065 | Zbl 1244.13008
[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). MR 2530766
[4] Behboodi, M., Shojaee, S. H.: Commutative local rings whose ideals are direct sums of cyclic modules. submitted.
[5] Cohen, I. S.: Commutative rings with restricted minimum condition. Duke Math. J. 17 (1950), 27–42. DOI 10.1215/S0012-7094-50-01704-2 | MR 0033276 | Zbl 0041.36408
[6] Cohen, I. S., Kaplansky, I.: Rings for which every module is a direct sum of cyclic modules. Math. Z. 54 (1951), 97–101. DOI 10.1007/BF01179851 | MR 0043073 | Zbl 0043.26702
[7] Kaplansky, I.: Elementary divisors and modules. Trans. Amer. Math. Soc. 66 (1949), 464–491. DOI 10.1090/S0002-9947-1949-0031470-3 | MR 0031470 | Zbl 0036.01903
[8] Köthe, G.: Verallgemeinerte abelsche gruppen mit hyperkomplexen operatorenring. Math. Z. 39 (1935), 31–44. DOI 10.1007/BF01201343 | MR 1545487
[9] Warfield, R. B. Jr., : A Krull–Schmidt theorem for infinite sums of modules. Proc. Amer. Math. Soc. 22 (1969), 460–465. DOI 10.1090/S0002-9939-1969-0242886-2 | MR 0242886 | Zbl 0176.31401
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