... Q = 1. For an integral domain R, p. dimr Q = 1 if and only if every divisible R-module is h-divisible [4, Th. VI.1.3). In particular if p. dimr Q = 1, every simple divisible R-module is a quotient of Q and every torsion simple divisible ...
... domain, and let S : Q be a torsion-free R-module of rank one. Then the following are equivalent: (1) Hom R (Q, Q/S) = Q. (2) Extol (Q, S) = 0. (3) S so Extr" (K, S). If any of these conditions hold, then Q/S is an indecomposable R ...
... Q. Write 2-1 = c/d with d 4 Q. If c < Q then c e Q C dr and so 2-1 & R, (1:2-1) = R, and our assertion holds trivially. Therefore assume that c 4 Q, as well. As R is a distinguished domain, (; ; d) does not consist entirely of zero ...
... domains Q,. Hence |Sww.lo, a = 2. Sw(W.Va) p p, o, < *X, solo, sco.1%. We return to formula (6.7). If v, belongs to C*(Q) s L*(Q) then by Fatou's lemma |S., slim in Sol, sco.; It follows that if such holds for all a then Rw belongs to W**(Q ...
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... (Q, X).S () L, for i = 1, ..., n. Then (0) CP1' C ... CP,' is a saturated chain of prime ideals of length n in L such ... domain A, then P = (q, X).A[X] is a prime ideal such that height P = height q + 1 and depth P = depth q. Note, on ...
... Q be a prime ideal in an integral extension domain B of a Noetherian domain A. Then height QBIX] = height Q, depth QBs X] = depth Q + 1, QBLX]^ A[X] = (Q (YA)A[X], and B and Bo are strong Sdomains. Proof. It will first be shown that B ...
... Q. For the smooth cone property the cones C. must vary smoothly from point to point. Muramatu [13, p. 328] uses a smooth cone condition in his approach to Sobolev and Besov spaces ... Q be a domain in 192 R. A. ADAMS AND J. J. F. FOURNIER.
... q is injective and B is a domain whose quotient field is an algebraic extension of the quotient field of A. Then B is esspec-residually finite. Proof. Since q is an ess-residually finite homomorphism, it suffices to check that the ...
domain of RP/Qp. If m € S(T) then there is a prime q in R with height q = m. and depth q = d. Proof. Since RP/QP = (R/Q) pyo, T = Dw, where D is an integral extension domain of R/Q and N is the multiplicatively closed subset R/Q — P/Q ...