Tertiary ideal
Encyclopedia
In mathematics
, a tertiary ideal is an (two-sided) ideal
in a (perhaps noncommutative) ring
that cannot be expressed as a nontrivial intersection of a right fractional ideal
with another ideal. Tertiary ideals generalize primary ideal
s to the case of noncommutative ring
s. Although primary decompositions do not exist in general for ideals in noncommutative rings, tertiary decompositions do, at least if the ring is Noetherian
.
Every primary ideal is tertiary. Tertiary ideals and primary ideals coincide for commutatitve rings. To any (two-sided) ideal, a tertiary ideal can be associated called the tertiary radical, defined as
Then t(I) always contains I.
If R is a (not necessarily commutative) Noetherian ring and I a right ideal in R, then I has a unique irredundant decomposition into tertiary ideals
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a tertiary ideal is an (two-sided) 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"....
in a (perhaps noncommutative) 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...
that cannot be expressed as a nontrivial intersection of a right fractional ideal
Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed...
with another ideal. Tertiary ideals generalize primary ideal
Primary ideal
In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n...
s to the case of noncommutative ring
Noncommutative ring
In mathematics, more specifically modern algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, if R is a noncommutative ring, there exists a and b in R with a·b ≠ b·a, and conversely.Noncommutative rings are ubiquitous in mathematics, and occur...
s. Although primary decompositions do not exist in general for ideals in noncommutative rings, tertiary decompositions do, at least if the ring is Noetherian
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
.
Every primary ideal is tertiary. Tertiary ideals and primary ideals coincide for commutatitve rings. To any (two-sided) ideal, a tertiary ideal can be associated called the tertiary radical, defined as
Then t(I) always contains I.
If R is a (not necessarily commutative) Noetherian ring and I a right ideal in R, then I has a unique irredundant decomposition into tertiary ideals
.