The Model Companion of the Theory of Commutative Rings without Nilpotent Elements

نویسنده

  • L. LIPSHITZ
چکیده

We show that the theory of commutative rings without nilpotent elements has a model companion. The model companion is decidable and is the model completion of the theory of commutative regular rings. Recall that a theory K is model-complete if for any model M of K, KuD(M) is complete, where D(M) denotes the diagram of M. A natural generalization of this notion is that of a model completion. We say that K' is a model completion of K if K' extends K and, for any model M of K, K'\JD(M) is consistent and complete (see [5]). For example the theory of algebraically closed fields is a model completion of the theory of fields and the theory of real closed fields is a model completion of the theory of ordered fields. A further generalization is the idea of a model companion. We say that K and K' are mutually model consistent if every model of K can be embedded in a model of K' and vice versa. K' is a model companion of K if K and K' are mutually model consistent and K' is model-complete. Model completions and model companions (when they exist) are unique. For this and other elementary properties see [5] and [6]. In everything that follows we shall use the word ring to mean ring with identity. We call a ring £ regular (in the sense of von Neumann) if for any x e R there exists y e R such that xyx=x. (A good general reference for the algebra relevant to this paper is [3].) Notice that in any commutative ring the set of idempotents forms a Boolean algebra under the operations eKJf=e+f—ef, eC\f=ef. Hence when we say that e is a subidempotent off we mean that ef=e (i.e. eC\f=e). e is a minimal idempotent if e/=/implies that/is either e or 0. We shall say that a quantifierfree formula y(alt ■ • • , an) holds on an idempotent e of a ring £ if the formula obtained from y> by multiplying every term in y by e holds in £. Received by the editors February 25, 1972 and, in revised form, July 28, 1972. AMS (MOS) subject classifications (1970). Primary 02H05, 02H99; Secondary 02G05, 02G20.

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تاریخ انتشار 2010