Artinian ideal

In abstract algebra, an Artinian ideal, named after Emil Artin, is encountered in ring theory, in particular, with polynomial rings.
Given a polynomial ring "R" = "k"["X"1, ... "X""n"] where "k" is some field, an Artinian ideal is an ideal "I" in "R" for which the Krull dimension of the quotient ring "R"/"I" is 0. Also, less precisely, one can think of an Artinian ideal as one that has at least each indeterminate in "R" raised to a power greater than 0 as a generator. 
If an ideal is not Artinian, one can take the Artinian closure of it as follows. First, take the least common multiple of the generators of the ideal. Second, add to the generating set of the ideal each indeterminate of the LCM with its power increased by 1 if the power is not 0 to begin with. An example is below.
Examples.
Let formula_1, and let formula_2 and formula_3. Here, formula_4 and formula_5 are Artinian ideals, but formula_6 is not because in formula_6, the indeterminate formula_8 does not appear alone to a power as a generator. 
To take the Artinian closure of formula_6, formula_10, we find the LCM of the generators of formula_6, which is formula_12. Then, we add the generators formula_13, and formula_14 to formula_6, and reduce. Thus, we have formula_16 which is Artinian.