We show that there is a one-to-one correspondence (up to isomorphism) between commutative rings with unity and metabelian commutative loops belonging to a particular finitely axiomatizable class. Based on this correspondence, it is proved that the sets of identically valid formulas and of finitely refutable formulas of a class of finite nonassociative commutative loops (and of many of its other subclasses) are recursively inseparable. It is also stated that nonassociative commutative free automorphic loops of any nilpotency class have an undecidable elementary theory.

中文翻译：

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交换环与乔丹环的对应关系

我们表明，在具有统一性的交换环和属于特定有限公理化类的 metabelian 交换环之间存在一对一的对应关系（直到同构）。基于这种对应关系，证明了一类有限非结合交换循环（及其许多其他子类）的相同有效公式集和有限可反驳公式集是递归不可分的。还指出任何幂零类的非结合交换自由自守循环都有一个不可判定的基本理论。