Chain Conditions On Annihilator Ideals
The lattice of ideals of the form where S is a subset of R comprise a complete lattice when partially ordered by inclusion. It is interesting to study rings for which this lattice (or its right counterpart) satisfy the ascending chain condition or descending chain condition.
Denote the lattice of left annihilator ideals of R as and the lattice of right annihilator ideals of R as . It is known that satisfies the A.C.C. if and only if satisfies the D.C.C., and symmetrically satisfies the A.C.C. if and only if satisfies the D.C.C. If either lattice has either of these chain conditions, then R has no infinite orthogonal sets of idempotents. (Anderson 1992, p.322) (Lam 1999)
If R is a ring for which satisfies the A.C.C. and RR has finite uniform dimension, then R is called a left Goldie ring. (Lam 1999)
Read more about this topic: Annihilator (ring Theory)
Famous quotes containing the words chain, conditions and/or ideals:
“It could not have come down to us so far,
Through the interstices of things ajar
On the long bead chain of repeated birth,
To be a bird while we are men on earth,”
—Robert Frost (1874–1963)
“The circuited city of the future will not be the huge hunk of concentrated real estate created by the railway. It will take on a totally new meaning under conditions of very rapid movement. It will be an information megalopolis.”
—Marshall McLuhan (1911–1980)
“We want our children to become warm, decent human beings who reach out generously to those in need. We hope they find values and ideals to give their lives purpose so they contribute to the world and make it a better place because they have lived in it. Intelligence, success, and high achievement are worthy goals, but they mean nothing if our children are not basically kind and loving people.”
—Neil Kurshan (20th century)