Let R be a ring with identity. By a central ideal of R we mean an ideal of R which can be generated by central elements of R. An ideal I of R is called hypercentral provided there exists a transfinite chain of ideals of R

0 = I₀1 α α +1 ρ = I,

where for each ordinal 0 ≤ α ≤ ρ the ideal I_α+1/I_α is a central ideal of the ring R/I_α and I_α = ∪_{0≤ β ≤ α} I_β for every limit ordinal 0 height of I. If M is a unitary right R-module and J an ideal of R then we set ann_M(J) = {m ∈ M | mJ = 0 }.

Our starting point is a theorem of Robinson proved in 1974 that states that if M is a Noetherian right R-module and
I a hypercentral ideal of R of height ω then there exists a positive integer n such that

ann_M(I) ∩ MIⁿ = 0. There is a dual result for Artinian modules proved by Newell in 1976. Robinson showed that his theorem does not extend to hypercentral ideals of height ω + 1 and Dark (1976) showed that the same was true for Newell's theorem. A good source of examples of hypercentral ideals is to be found in group rings. Roseblade (1971) proved that every ideal of the integral group ring ZG is hypercentral if and only if the group G is hypercentral. This leads to a consideration of the Artin Rees Property and to certain chain conditions in groups and in rings.