Geodesic
Generalized bases in certain rings of holomorphic functions
Izvestiya. Mathematics , Tome 6 (1972) no. 3, pp. 564-578

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Conditions are advanced in the paper which, imposed on a ring of holomorphic functions of several variables and a finite collection of functions in that ring, are sufficient for the whole ring to be generated as the closed ideal spanned by the functions in the collection. 
@article{IM2_1972_6_3_a7,
     author = {D. I. Gurevich},
     title = {Generalized bases in certain rings of holomorphic functions},
     journal = {Izvestiya. Mathematics },
     pages = {564--578},
     publisher = {mathdoc},
     volume = {6},
     number = {3},
     year = {1972},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1972_6_3_a7/}
}
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D. I. Gurevich. Generalized bases in certain rings of holomorphic functions. Izvestiya. Mathematics , Tome 6 (1972) no. 3, pp. 564-578. http://geodesic.mathdoc.fr/item/IM2_1972_6_3_a7/