Artinian ideal
Encyclopedia
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[X1, ... Xn] 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.
, and let 
and 
. Here, 
and 
are Artinian ideals, but 
is not because in 
, the indeterminate 
does not appear alone to a power as a generator.
To take the Artinian closure of
, 
, we find the LCM of the generators of 
, which is 
. Then, we add the generators 
, and 
to 
, and reduce. Thus, we have 
which is Artinian.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, an Artinian ideal, named after Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
, is encountered in ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
theory, in particular, with polynomial rings.
Given a polynomial ring R = k[X1, ... Xn] where k is some field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, an Artinian ideal is an ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
I in R for which the Krull dimension
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
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, and let and . Here, and are Artinian ideals, but is not because in , the indeterminate does not appear alone to a power as a generator.To take the Artinian closure of
, , we find the LCM of the generators of , which is . Then, we add the generators , and to , and reduce. Thus, we have which is Artinian.