What is a word to describe something which cannot be broken down any further?

I don't want to use the word elemental, nor axiomatic, as both terms are already defined and in popular use.

Anyway, the way I want to use the term is not to describe the part, or type of part itself, but the quality that the part possesses; 'Something with which cannot be broken down any further possesses the quality x', where x is the word I seek, in the same way an axiom has the quality of being axiomatic.

Solution 1:

I think that the word atomic, which is often otherwise used in a physical sense, should be used here.

atomic a·tom·ic /əˈtämik/ adjective

• of or relating to an atom or atoms. "the atomic nucleus"

• Chemistry (of a substance) consisting of uncombined atoms rather than molecules. "atomic hydrogen"

• of or forming a single irreducible unit or component in a larger system.

–Oxford Dictionaries

Solution 2:

I'd go for irreducible, "not reducible; incapable of being reduced or of being diminished or simplified further".