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72(1-2):167-195, 1990. [9] B. Stenstrom. Rings and modules of quotients, volume No. 237 of Lecture Notes in Mathematics. Springer Verlag, 1971. 37 [10] M. E. Sweedler. Hopf Algebras. W. A. Benjamin, New York, 1969. [11] M. Takeuchi. Formal schemes over fields. Comm. Algebra, 5:1483-1528, 1977. [12] M. Takeuchi. Morita theorems for categories of comodules. J. Fac. Sci. Univ. Tokyo, 24:629-644, 1977. [13] M. Tkkeuchi. A note on geometrically reductive groups. J. Fac. , Univ. Tokyo, Sect. 1, 20(3):384-396, 1973.

Evidently, an integral domain R with quotient field K is a pseudovaluation domain if and only if R C K is a strong extension. The following is an example of of a strong overring extension R C T of domains for which neither R nor T is quasilocal, (and hence neither R nor T is a pseudo-valuation domain. 1. 1];. Let L be the quotient field of Z[X], and V — L + XL[[X}] (observe that V is a valuation domain with maximal ideal XL[[X]}). SetR=Z + XL[[X\\ and T = Z[X] + XL[[X]]. Then RcT is a strong overring extension with the stated properties.

Lemma. Let M G Mc, N £ CM. For every W £ R-Mod there exist two canonical R-linear maps j w • W ® (MCN) —> (W ® M)CN, fiw • (McN) ® W —> Mc{N ® W). Moreover, the following are equivalent (1) MCN is W-pure in M ® N'. (2) 7n/ is an isomorphism. (3) Hw is an isomorphism. Remark:. 3. fi-Mod and consider the following diagram, 0[r]W <8> {McN)[r]@— >[d\ M)cN[r](W ® in which the second row is exact by definition. R-linear map jw exists and render the diagram commutative.

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