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)$$...

It is shown that the right-adequate duo ring with a nonzero Jacobson radical has stable range one. established class of complete matrices order 2 over elementary divisors Some classes unit-regular and clean are investigated.

A ring [Formula: see text] is said to be clean if each element of can written as the sum a unit and an idempotent. weakly either or difference idempotent, feebly every text], where are orthogonal idempotents. Clearly, rings both them clean. In recent paper (J. Algebra Appl. 17 (2018) 1850111 (5 pp.)), McGoven characterized when group clean, distinct primes. this paper, we consider more general ...

Cryogenic targets, thin plastic shells surrounding thick cryogenic deuterium layers, are used to achieve nuclear fusion. These targets are irradiated by lasers causing the targets to implode and create fusion reactions by forcing the deuterium inside to fuse together. The cryogenic targets are pressurized with deuterium gas, and then the targets are chilled to about 20K to freeze an ice layer o...

Let I be a monomial ideal of the polynomial ring S = K[x1, . . . , x4] over a field K. Then S/I is sequentially Cohen-Macaulay if and only if S/I is pretty clean. In particular, if S/I is sequentially Cohen-Macaulay then I is a Stanley ideal.