Are this below monomial or polynomial?
$2x^{-5}$
$5^{-2}x^{-123}$
etc.
I have read the monomial. I wonder if one of the monomial rule says that the monomial must have the exponent of variable is zero or positive integer. In the case where the exponent of a variable is negative it be monomial or polynomial.
I would say that it's neither. Monomials are typically understood to be natural powers of the variable(s), so negative integers are not included. Polynomials are finite sums of monomials.
As mentioned in a comment, the concept of polynomials has been extended to negative powers. The concept is called Laurent polynomial, so it makes sense to call your objects Laurent monomials.
However, I think it would be best in a typical context to use a less specific name. Since "Laurent monomial" will not be recognized and easily parsed by many, you might be better off calling them "terms" or some such thing, depending on context.