Set of all non negative integers in the language of rings

Question

If the language I am working with is $(\mathbb{Z},+,\cdot,0,1)$, then can the set of non-negative integers be expressed as a definable set? I understand that if I add a binary symbol $\leq$, I have a formula but what about when the binary symbol is not given, i.e. how do I write the formula in this language? Thank you for any suggestions or hints.

Answer 1

You could leverage Lagrange's four-square Theorem: $$p(x) := \exists a,b,c,d. ~~ x = a \cdot a + b \cdot b + c \cdot c + d \cdot d$$