A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal...

the length of equal minimal and maximal blocks has eected on logarithm-scale logarithm against sequential function on variance and bias of de-trended uctuation analysis, by using quasi monte carlo(qmc) simulation and cholesky decompositions, minimal block couple and maximal are founded which are minimum the summation of mean error square in horest power.

In every dense poset P every maximal antichain S may be partitioned into disjoint subsets S1 and S2 , such that the union of the upset of S1 with the downset of S2 yields the entire poset: U(S1) [ D(S2) = P . To nd a similar splitting of maximal antichains in posets is NP{hard in general.

Let ` be a fixed rational prime number. Consider function fields K|k over algebraically closed fields k of characteristic 6= `. For each such a function field K|k, let Π K := Gal(K ′′|K) be the Galois group of a maximal pro-` abelian-by-central Galois extension K ′′|K, and ΠK = Gal(K ′|K) be the Galois group of the maximal pro-` abelian sub-extension K ′|K of K ′′|K. At the beginning of the 199...