ON HOPF GALOIS HIRATA EXTENSIONS
Let H be a finite-dimensional Hopf algebra over a field K, H* the dual Hopf algebra of H, and B a right H*-Galois and Hirata separable extension of BH. Then B is characterized in terms of the commutator subring VB(BH) of BH in B and the smash product VB(BH)#H. A sufficient condition is also given for B to be an H*-Galois Azumaya extension of BH.
NOTE ON SEPARABILITY OF ENDOMORPHISM RINGS
It is well known that the endomorphism ring [)=End (RE) of a finitely generated projective module E over a commutative ring R is a separable Rュ algebra with R/αits center, where αis the annihilator ideal of E in R. Then, there comes out a problem whether or not this theorem holds in case R is non commutative and E is an R・R・.bimodule. Partially, an a伍rmative answer was given in Theorem 1 [11]. In this paper we give some su伍cient conュ ditions for [) to be separable over R in the case where M is an R-Rュ bimodule. In ~3 we give our main results. For example, if RMR is left R-finitely generated projective and R is isomorphic to a direct summand of a 五nite direct sum of copies of M as R-R-module, [) is separable over R (Theorem 6). This is a generalization of the well known result in com・ mutative case because every 五nitely generated projective module E over a commutative ring R is an R/α-progenerator. Furthermore, we obtain that for a ring extension A IF such that A is left r -progenerator, [) = End (rA) is separable over A if and only if r is a r-r-direct summand of A (Theorem 7). The 白・st two sections are devoted to the preperations for ~ 3. And we introduce the notions of M-semisimplisity, M-separability and centrally Mュ separability. These are equal to the notions of semisimple, separable and H幽 separable extensions respectively, in case M=SコR. In ~4 we give some commutor theory for some separable extensions.
Net Spaces in Categorical Topology a
In the paper we introduce the notion of S‐nets, S a nonempty construct, as a generalization of the usual nets. S‐net spaces and continuous maps between them are then defined in a natural way. For S‐net spaces we introduce a number of axioms and study the categories obtained. We find some topological and Cartesian closed categories of S‐net spaces and describe their relations to certain known topological categories.