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In the signature $(+,-,*,0,1)$, rings can be axiomatized by $\forall$-sentences. What about in the signature $(+,*,0,1)$? I believe they cannot, because you need an existential quantifier to state that every element has an additive inverse. But perhaps they can still be axiomatized by using another set of $\forall$-sentences. So, my question is, can the theory of rings in the signature $(+,*,0,1)$ be axiomatized by only $\forall$-sentences?

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  • $\begingroup$ See my comment here for some conceptual background on such matters. $\endgroup$ Commented Apr 20, 2022 at 23:34

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No, a class of algebras axiomatized by universal sentences is closed under subalgebras. But the subalgebra $\mathbb{N}$ in the signature $(+,*,0,1)$ of the ring $\mathbb{Z}$ is not a ring.

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