Homomorphic Images of an Infinite Product of Zero-dimensional Rings
نویسندگان
چکیده
Let R = Q a2A R a be an innnite product of zero-dimensional chained rings. It is known that R is either zero-dimensional or innnite-dimensional. We prove that a nite-dimensional homomorphic image of R is of dimension at most one. If each R a is a PIR and if R is innnite-dimensional, then R admits one-dimensional homomorphic images. However, without the PIR hypothesis on the rings R a , we present examples to show that R may be innnite-dimensional while each nite-dimensional homomorphic image of R is zero-dimensional. We prove that a prime ideal of R of positive height is of inn-nite height, and we give conditions for an innnite product of zero-dimensional local rings to admit a one-dimensional local domain as a homomorphic image.
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