A Note on Unions of Ideals and Cosets of Ideals
نویسنده
چکیده
The paper gives a reenement to a well known and easy-to-prove result of basic ideal theory: If an ideal I is contained in the union of a given nite collection of ideals, with at most two of them not prime, I must be contained in one of them. The proposition, stated in this way, leaves out the special structure of the ring under consideration. We show that the lower bound implicit in the proposition can be replaced by the minimum over cardinalities of residue elds of the given ring. MSC 1991: 13A15 Closely related to the fact that an ideal can not be contained in the union of two arbitrary and a nite number of prime ideals unless it is contained in one of them (cf. 1], (1.B)), a vector space over a eld k can not be covered by a nite collection of proper subspaces unless their number exceeds the cardinality of k (cf..2], p.23. A similar proposition holds for residue classes of subspaces.). Since a nontrivial eld contains at least two elements, the aforesaid might suggest that the eld case is inherent in the general case treated by the above quoted proposition. And indeed, the basic results of this paper are
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تاریخ انتشار 1995