Equational classes and elementary sentences

[dyunov]

The following ring classes also seem to be equational/formed by elementary sentences, but are not marked yet. Note that 0, 1 and inverses of elements are described using elementary sentences that only use quantifiers and conjunction (and thus are suitable for equational classes): $a+x=x$; $\forall x$ $ax=xa=x$; $ax = xa = u \wedge \forall z$ $uz=zu=z$.

• [ ] $I_0$. Elementary. Equivalent to "Every principal right ideal not contained in $J(R)$ contains a nonzero idempotent." Then, $aR$ is in $J(R)$ whenever $\forall x, y$ $1+xay$ is invertible.
• [ ] Abelian. Elementary.
• [ ] L/R cohopfian. Elementary.
• [ ] compressible. Elementary.
• [ ] Dedekind finite. Elementary.
• [ ] directly irreducible. Elementary.
• [ ] division ring. Elementary.
• [ ] domain. Elementary.
• [ ] L/R duo. Equational. Equivalent to "All principal right ideals are two-sided". Then the formula is $\forall a,b$ $ba = af(a,b)$.
• [ ] GCD domain. Elementary. The GCD map is encoded by $\forall a,b \exists c \forall d,e \exists f$ $ad + be = cf$. Since "domain" is elementary, so is this. Equational (Not closed under products.)
• [ ] L/R Ore ring. Elementary. $\forall a,s (s \text{ is right regular} \Rightarrow \exists u,v (u \text{ is right regular} \wedge au = sv))$.
• [ ] prime ring. Elementary. $\forall a,b (a = 0 \vee b = 0 \vee \exists r,s (arbs \not =0))$.
• [ ] reversible. Elementary.
• [ ] L/R Rickart. Elementary. $\forall x \exists e \forall a \exists s (xa = 0 \Rightarrow a = es \wedge e^2 = e)$.
• [ ] semicommutative. Elementary.
• [ ] semiprimitive. Elementary. $\forall r (\forall x, y (1 + xry \text{ invertible}) \Rightarrow r = 0)$.
• [ ] stable range 1. Equational. $\forall a,x \exists y, u (a+y(1-xa)=u^{-1})$.
• [ ] strongly connected. Elementary.
• [ ] symmetric. Elementary.
• [ ] L/R UGP ring. Elementary.

[rschwiebert]

GCD domain can't be equational: the product of two such rings is not a GCD domain. But maybe the "GCD" part without "domain" might be equational?

Can't remember if Jose and I checked stable range 1. I think the conditional bit of the definition given currently doesn't qualify. Are you using an alternative criterion?

[dyunov]

@rschwiebert Yes, "GCD ring" would be equational :-) Then GCD domain is elementary. I avoided the conditional in "stable range 1" by "solving for $b$".

[JoseBrox]

* [ ]  GCD domain. _Elementary._ The GCD map is encoded by ∀a,b∃c∀d,e∃f ad+be=cf. 

I'm not sure about this: this is enconding the existence of a common divisor, but it doesn't enforce its uniqueness (it doesn't really matter if we ask the ring to be a domain and we are satisfied with "elementariness").

[dyunov]

@JoseBrox Please excuse me, I don't understand the parenthesized part of your message, as the property in DaRT already includes "domain". Did you mean to add this as a separate entry akin to "Ore domain/ring" or "PID/ring"? This one would be slightly weaker than Bezout, right?