Atomic number consistent with the modern practice of ordering the elements by proton number, "Z", but that number was not known or suspected at the time. A simple numbering based on periodic table position was never entirely satisfactory, however. Besides the case of iodine and tellurium, later several other pairs of elements (such as argon and potassium, cobalt and nickel) were known to have nearly identical or reversed atomic weights, thus requiring their placement in the periodic table to be determined by their chemical properties. However the gradual identification of more and more chemically similar lanthanide elements, whose atomic number was not
Atomic domain ring can be "decomposed" without requiring uniqueness. The concept of an atomic domain addresses this. Let "R" be an integral domain. If every non-zero non-unit "x" of "R" can be written as a product of irreducible elements, "R" is referred to as an atomic domain. (The product is necessarily finite, since infinite products are not defined in ring theory. Such a product is allowed to involve the same irreducible element more than once as a factor.) Any such expression is called a factorization of "x". In an atomic domain, it is possible that different factorizations of the same element "x"