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)
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