Abstract
Let R be a commutative ring with the unit element. It is shown that an ideal I in R is pure if and only if Ann(f) + I = R for all f ∈ I. If J is the trace of a projective R-module M, we prove that J is generated by the “coordinates” of M and JM = M. These lead to a few new results and alternative proofs for some known results.
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References
Aghajani, M., Tarizadeh, A.: Characterizations of Gelfand rings specially clean rings and their dual rings. Results Math. 75(3), 125 (2020)
Auslander, M., Goldman, O.: Maximal orders. Trans. Am. Math. Soc. 97, 1–24 (1960)
Beckwith, H.: On trace for modules. Nagoya Math. J. 40, 121–131 (1970)
Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press (1956)
Dao, H., Kobayashi, T., Takahashi, R.: Trace ideals of canonical modules, annihilators of Ext modules, and classes of rings close to being Gorenstein. J. Pure Appl. Algebra 225(9), 106655 (2021)
de Jong, A.J., et al.: The Stacks Project. see http://stacks.math.columbia.edu (2021)
Herbera, D., Prihoda, P.: Reconstructing projective modules from its trace ideal. J. Algebra 416, 25–57 (2014)
Herzog, J., Hibi, T., Stamate, D. I.: The trace of the canonical module. Isr. J. Math. 233, 133–165 (2019)
Johndrup, S., Trosborg, P. J.: A remark on pure ideals and projective modules. Math. Scand. 35, 16–20 (1974)
Kaplansky, I.: Projective modules. Ann. Math. 68, 372–377 (1958)
Lindo, H.: Trace ideals and centers of endomorphism rings of modules over commutative rings. J. Algebra 482, 102–130 (2017)
Matsumura, H.: Commutative Ring Theory. Cambridge University Press (1989)
Northcott, D. G.: Multilinear Algebra. Cambridge University Press (1984)
Tarizadeh, A., Aghajani, M.: Structural results on harmonic rings and lessened rings. Beitr. Algebra Geom. https://doi.org/10.1007/s13366-020-00556-x (2021)
Tarizadeh, A., Aghajani, M.: On purely-prime ideals with applications. Commun. Algebra 49(2), 824–835 (2021)
Tarizadeh, A.: Notes on finitely generated flat modules. Bull. Korean Math. Soc. 57(2), 419–427 (2020)
Vasconcelos, W.: On finitely generated flat modules. Trans. Am. Math. Soc. 138, 505–512 (1969)
Vasconcelos, W.: Finiteness in projective ideals. J. Algebra 25, 269–278 (1973)
Whitehead, J. M.: Projective modules and their trace ideals. Commun. Algebra 8(19), 1873–1901 (1980)
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The author would like to give heartfelt thanks to the referee for very careful reading of the paper and for his/her excellent comments and modifications which significantly improved the paper.
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Tarizadeh, A. Some Results on Pure Ideals and Trace Ideals of Projective Modules. Acta Math Vietnam 47, 475–481 (2022). https://doi.org/10.1007/s40306-021-00451-0
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DOI: https://doi.org/10.1007/s40306-021-00451-0