In this paper, a combination of algebraic and topological methods are applied to obtain
new and structural results on harmonic rings. Especially, it is shown that if a Gelfand
ring A modulo its Jacobson radical is a zero dimensional ring, then A is a clean ring.
It is also proved that, for a given Gelfand ring A, then the retraction map Spec(A) →
Max(A) is flat continuous if and only if A modulo its Jacobson radical is a zero
dimensional ring. Dually, it is proved that for a given mp-ring A, then the retraction
map Spec(A) → Min(A) is Zariski continuous if and only if Min(A) is Zariski
compact. New criteria for zero dimensional rings, mp-rings and Gelfand rings are
given. The new notion of lessened ring is introduced and studied which generalizes
“reduced ring” notion. Especially, a technical result is obtained which states that the
product of a family of rings is a lessened ring if and only if each factor is a lessened
ring. As another result in this spirit, the structure of locally lessened mp-rings is also
characterized. Finally, it is characterized that a given ring A is a finite product of
lessened quasi-prime rings if and only if Ker πp is a finitely generated and idempotent
ideal for all p ∈ Min(A).