Commutative Rings with Finite Quotient Fields

We consider the class of all commutative reduced rings for which there exists a finite subset T ⊂ A such that all projections on quotients by prime ideals of A are surjective when restricted to T . A complete structure theorem is given for this class of rings, and it is studied its relation with other finiteness conditions on the quotients of a ring over its prime ideals. Introduction Our aim is to study the structure of commutative rings that satisfy suitable finiteness conditions on its quotients by prime ideals. If A is a commutative ring and A/p represents the quotient ring of A by a prime ideal p, we will be interested in the following conditions on A: (1) All A/p are finite. (2) The cardinal of all A/p is bounded by some n. (3) There are x1, . . . , xn ∈ A such that A/p = {x1 + p, . . . , xn + p} for any prime ideal p of A. Clearly, 3 ⇒ 2 ⇒ 1 and at the end of this article examples can be found showing that no converse is true. Also, observe that the three conditions are the same for A as for the reduced ring A/N(A), so we may restrict ourselves to reduced rings, that is, to commutative rings without nilpotent elements. One main result in this paper is that a complete structure theorem can be given for reduced rings satisfying condition (3). Namely, we associate to each such a ring A a tuple (K1, B1, . . . ,Kn, Bn), where the Ki’s are non isomorphic finite fields and the Bi’s are Boolean rings, in such a way that two of these rings are isomorphic if and only if the associated tuples are equal, up to isomorphism and order. A reduced ring satisfying just condition (1) must be Von Neumann regular (absolutely flat, in other terminology), as it is any reduced ring in which all prime ideals are maximal [2, Theorem 1.16]. In this sense, our work continues that of N. Popescu and C. Vraciu in [3], where the structure of commutative Von Neumann regular rings is studied. Here it is shown that imposing these kind of finiteness conditions, strong structure results can be given. Let us note that C. Vraciu had already found in [4], in other direction, the following result concerning rings satisfying condition (1): 2000 Mathematics Subject Classification. 19AXX, 16E50. Author supported by FPU grant of SEEU-MECD, Spain. 1