The space of maximal ideals is studied on semiprimitive rings and reduced rings, and the relation between topological properties of Max(R) and algebric properties of the ring R are investigated. The socle of semiprimitive rings is characterized homologically, and it is shown that the socle is a direct sum of its localizations with respect to isolated maximal ideals. We observe that the Goldie dimension of a semiprimitive ring R is equal to the Suslin number of Max(R).

Let be a proper ideal of a commutative Noetherian ring R of prime characteristic p and let Q() be the smallest positive integer m such that ${{(}^F)}^{\left[p^m\right]}{=}^{\left[p^m\right]}$, where ${}^F$ is the Frobenius closure of . This paper is concerned with the question whether the set $Q{(}^{\left[p^m\right]}):m\in \mathbb{N}₀$ is bounded. We give an affirmative answer in the case that the ideal is generated by an u.s.d-sequence c₁,..., cₙ for R such that (i) the map $R/{\sum }_{j=1}^{n}R{c}_j\to R/{\sum }_{j=1}^{n}Rc{²}_j$ induced by multiplication by c₁...cₙ is an R-monomorphism; (ii) for all $\in ass(c{₁}^j,...,c{ₙ}^j)$, c₁/1,..., cₙ/1 is a $R_{}$-filter regular sequence...