PERFECT ROLE OF ATOM IN WEAK IDEMPOTENT RINGS

Abstract

In a commutative weak idempotent ring R with unity, we prove that bR = {0, b}
or {0, b, n, b + n} for idempotent and nilpotent atoms b and n of R respectively provided that any
two nilpotent atoms have an upper bound in R. Further, we prove that the subgroup generated by
{ni}i∈I in R is a lattice, where {ni}i∈I is the collection of nilpotent atoms of R corresponding to the
idempotent atoms {bi}i∈I of R. We also prove that R is atomic if and only if RB is atomic provided
that the set of all nilpotent elements of R is nil-free and any two nilpotent atoms have an upper
bound in R. Finally, we state and prove the direct product decomposition theorem of R.