 Given any subset "F" = {"f""j"}"j" ∈ "J" of "R" (where "J" is some index set), the ideal "generated by F" is the smallest ideal that contains "F".Equivalently, it is given by finite linear combinations

If "F" consists of a single element "r", the ideal generated by "F" consists of the multiples of "r", i.e., the elements of the form "rs" for arbitrary elements "s".Such an ideal is called a principal ideal.