In this paper, a combination of algebraic and topological methods are applied to obtain new structural results on harmonic rings. Especially, it is shown that if Gelfand ring A modulo its Jacobson radical zero dimensional ring, then clean ring. It also proved that, for given A, the retraction map $${\text {Spec}}(A)\rightarrow {\text {Max}}(A)$$ flat continuous only Dually, mp-ring {Min}}(A)$$...

We revisit and further explore the celebrated Combinatorial Nullstellensätze of N. Alon in several different directions. Terminology: Throughout this note, a “ring” is a commutative ring with multiplicative identity. A “domain” is a ring R such that for all a, b ∈ R \ {0}, ab ̸= 0. A ring R is “reduced” if for all x ∈ R and n ∈ Z, if x = 0 then x = 0.

The Src family of protein tyrosine kinases have been implicated as important regulators of cellular proliferation, differentiation and function. In order to understand further the role of Src family kinases, we have generated loss-of-function mutations in Src64, one of two Src family kinases known in Drosophila melanogaster. Animals with reduced Src64 function develop normally and are fully via...