The Logical Complexity of Finitely Generated Commutative Rings

Abstract

We characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with (N, +, ×) if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in Z. Notably, by constructing a nontrivial derivation on a nonstandard model of arithmetic we show that the ring of dual numbers over Z is not bi-interpretable with N.

Uniform Definability of Integers in Reduced Indecomposable Polynomial Rings

“…This will be used in the proof of the following result, which shares the same spirit of Proposition 4.2, but concerning indecomposability: , yielding e 0 = e 2 0 , and in particular (1 -2e 0 )gx = (gx) 2 . Since (1 -2e 0 ) 2 = 1, it follows that (1 -2e 0 )gx = [(1 -2e 0 )gx] 2 . Thus (1 -2e 0 )gx = [(1 -2e 0 )g] n x n for all n ≥ 1, that is, (1 -2e 0 )gx is infinitely divisible by x, and so necessarily (1 -2e 0 )gx = 0.…”

“…From the very definition of polynomials and their multiplication, it follows that 0 is the only polynomial infinitely divisible by x. This will be used in the proof of the following result, which shares the same spirit of Proposition 4.2, but concerning indecomposability: , yielding e 0 = e 2 0 , and in particular (1 -2e 0 )gx = (gx) 2 . Since (1 -2e 0 ) 2 = 1, it follows that (1 -2e 0 )gx = [(1 -2e 0 )gx] 2 .…”

“…For instance, Julia Robinson showed that Z is firstorder definable in Q [21]. Concerning negative results, it was recently proved [2,Lemma 4.7] that the direct product of two infinite finitely generated rings is not bi-interpretable with Z; the proof of this result can be easily seen to contain, in its essence, the fact that Z is not definable in Z × Z, for instance.…”

Finite axiomatizability for profinite groups

“…Here SL 1 n (Z p ) denotes the principal congruence subgroup modulo p in SL n (Z p ). The proof for SL 1 n (Z p ) uses both Theorem 1.6 and Theorem 1.7, which can be applied to the upper unitriangular group (when n 3). The extension to SL n (Z p ) depends on Theorem 3.1, proved in §3, which establishes some sufficient conditions for a finite extension of an FA group to be FA.…”

Characterizing finitely generated fields by a single field axiom

“…One way of making this precise (cf. [AKNS,Lemma 2.17]) is the following statement. (See [Sc,Section 2] or [AKNS,Section 2] for a discussion of the notion of bi-interpretability.)…”