T-torsion theories and central localizations

Abstract

We use the notion of a critical right ideal as developed by Lambek and Michler [9] as a generalization of prime ideals in a commutative ring. Call a right ideal P of the ring R critical if for every H properly containing P, Hom(R/H, E(r/P)) = 0 where E(R/P) is the injective hull of R/P (Gordon and Robson call R/P monoform). In the first section we look at T-torsion theories (T in the sense of Goldman [4]). Let S be a ring of quotients with respect to some T-torsion theory. Then there is a natural bijection between critical right ideals of S and closed critical right ideals of R (P closed means R/P torsion-free). We also prove that this torsion theory is cogenerated by a family of critical right ideals. A corollary of this gives a generalization of a result of Richman [12]. Namely, we show that if R is prime Goldie and S is a left flat, epimorphic ring extension of R, then S = n RP where P runs over some family of critical right ideals. Here R, denotes the ring of quotients with respect to the torsion theor! cogenerated by E(R/P). In the second section we look at localizations of R at prime ideals of the Center R = -4. In the most general case, let R be an arbitrary ring (with identity). Then w. dim R = sup w. dim R./l , where ;K runs through the maximal ideals of .-I. (w. dim R is weak or flat dimension of R.) \Ve then investigate the case when R either is (i) finitely generated over A (as a module) or (ii) R is Xoetherian and is integral over A. If R satisfies either of these conditions, then there is a canonical bijection between (two-sided) prime ideals of R and the equivalence classes of critical right ideals (PI is equivalent to P2 if E(R/P,) N E(R/P,)). Theorem 2.14 and Corollary 2.15 give the main results of this paper, which are as follows: If R satisfies either (i) or (ii), then there exists a family F of semiprime ideals such that: (I) for each N E 7, localization at N is T; NR, = Jacobson radical of R, ; and R,/NR, is Artinian; and (2) an R module M is flat iff MA, is R,\flat for all NE .F.