5

Motivation:

According to this, the minimum number of axioms required to define a group is one.

What can we say about rings (that are not necessarily commutative nor do they have to have a one)?

The Question:

What is the minimum number of axioms necessary to define a (not necessarily commutative) ring (that doesn't have to have a one)?

Details:

This is in the spirit of the link above: you can have as many operations as you like (e.g., $*$ and $(\cdot)'$ for an abelian group, under the axiom

$$\forall x\forall y\forall z, ((x * y) * z) * (x * z)' = y,$$

as cited), as long as you use just one axiom. (I guess the closure of each operation is given.)

I'm afraid I can't define this more rigorously.

Thoughts:

My guess is that two or three might suffice. Why? I don't know.

I have no experience in this sort of thing, so please pitch your answers at an introductory level, perhaps for a bright undergraduate, if possible.

This seems to be the domain of . I have included the tag , too, because it might attract people who are interested.

Shaun
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    Caveat: The mentioned "single axiom" for defining groups is not enough, since the empty set also satisfies the axiom. Usually you see the definition 1 = x/x for "any" x then, which is not sufficient of course. – Martin Brandenburg Dec 31 '23 at 13:51
  • Perhaps $e$ is a nullary operation, @MartinBrandenburg . . . I don't know. – Shaun Jan 01 '24 at 13:55

1 Answers1

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The class of nonunital rings in the language of (binary) multiplication, $xy$, and (binary) subtraction, $x-y$, can be axiomatized by a single identity. For this, see Theorem 4 of

George McNulty
Minimum Bases for Equational Theories of Groups and Rings:
the work of Alfred Tarski and Thomas Green,
Annals of Pure and Applied Logic, 2004.

In fact, Theorem 4 is much stronger than is needed for the conclusion that I have drawn from it. Let me include the theorem here. For this, let $T$ denote an equational theory of rings in a language with $n$ operation symbols. For example, $T$ might be a theory of rings in the language of two binary operations $xy, x-y$, or $T$ might be a theory of rings in the language of two binary operations and one unary operation $xy, x+y, -x$. Let $\mu T$ denote the least size of a subset $A\subseteq T$ that axiomatizes $T$ (i.e., $A$ is a subset of $T$ and $A$ and $T$ have the same equational consequences). Finally, call $R$ a 'zero ring' if its multiplication is constant.

Theorem 4. Assume that $T$ is finitely axiomatizable (i.e. $\mu T$ is finite).

  • If $T$ is a theory of nonunital rings and $T$ has a model that is a zero ring of more than one element, then $\mu T = \max(1,n-1)$.
  • If $T$ is a theory of nonunital rings and $T$ has no model that is a zero ring of more than one element, then $\mu T = 1$.
  • If $T$ is a theory of unital rings, then $\mu T = 1$.

    • The theory $T$ of ALL nonunital rings has a model that is a zero ring of more than one element, so if this theory is presented in the language of two binary operations $xy$ and $x-y$, then $\mu T=1$ and the theory has a set of axioms consisting of a single identity. On the other hand, of the theory is presented in a language of two binary operations $xy, x+y$ and one unary operation $-x$, then $\mu T=2$, so in this language $T$ has a set of axioms consisting of two identities, but no set of axioms consisting of one identity.

    Theorem 4 need not be applied to the class of ALL rings, but could be applied to a subvariety, like the class of commutative rings. The theorem gives the same results: if the theory $T$ of ALL nonunital commutative rings is presented in the language of two binary operations $xy$ and $x-y$, then $\mu T=1$. If the theory is presented in a language of three operations $xy, x+y, -x$, then $\mu T=2$.

    Keith Kearnes
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