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:
“To avoid tripping on the chain of the past, you have to pick it up and wind it about you.”
—Mason Cooley (b. 1927)
“We have got onto slippery ice where there is no friction and so in a certain sense the conditions are ideal, but also, just because of that, we are unable to walk. We want to walk so we need friction. Back to the rough ground!”
—Ludwig Wittgenstein (1889–1951)
“The old ideals are dead as nails—nothing there. It seems to me there remains only this perfect union with a woman—sort of ultimate marriage—and there isn’t anything else.”
—D.H. (David Herbert)