Some conditions on commutative semiprime rings

On commutative P. P. rings

“…Furthermore it can be shown as a corollary of Theorem 3.1 of Mewborn [12], that conditions (4) and (5) are equivalent for a semiprime ring R if and only if Minp R is compact.…”

“…(1) implies (2). Q is a flat i?-module as Minp R is compact [12,Thm 3.1]. As R is also semiprime it is possible to use Theorem 1.6 of [3].…”

“…Thus 1 e JK and 1 = Σ?^i ^ = Σ?=i ^(MΓ 1 ) where XiβJ, d { is a nonzero divisor of i?. Now by [12,Lemma 1.3] there exist biβ R and a nonzero divisor d e R such that a { d~ι -b^1 and so 1 = Σ^δiCZ" Proof. (1) implies (2).…”

Prime ideals and sheaf representation of a pseudo symmetric ring

“…Proof, (e) =► (d) always, for if Q e fp| = hull P then P Ç Q and 0(P) C Q. Mewborn [13] has obtained a characterization of a commutative ring with identity whose minimal prime ideal space is compact, generalizing the result due to Henriksen and Jerison [6]. Our aim here is to obtain a similar characterization for the noncommutative case.…”

Valuations and Rings of Quotients

“…Among all of the regular rings containing the ring R in a given larger ring, there is a unique minimal regular ring containing R (cf. [6]). Thus Theorem 2.3 tells us that the valuations on any ring of quotients of the ring R are extensions of valuations on the minimal regular ring of quotients of R.…”