Compact rings

Throughout this abstract, R denotes a compact (Hausdorff) topological ring. The authors extend to ring topologies on R which are totally bounded or even pseudocompact; a principal tool is the Bohr compactification of a topological ring. They show inter alia: If the Jacobson radical J R of R satisfies w R/J R > ω then there is a pseudocompact ring topology on R strictly finer than ; if in addition w R = w R/J R = α with cf α > ω then there are exactly 2 2 R -many such topologies. The ring R is said to be a van der Waerden ring if is the only totally bounded ring topology on R. Theorem 4.13 asserts that if R is semisimple, then R is a van der Waerden ring if and only if in the Kaplansky representation R = n<ω R n α n of R as a product of full matrix rings over finite fields each α n is finite. Other classes of van der Waerden rings are constructed, and it is shown that there are non-compact totally bounded rings S such that is the only totally bounded ring topology on S.

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“…[44,Theorem 8]). In particular, the valuation ring of a proper real valuation of a locally compact division ring is Noetherian.…”

Section: Lemma Let R Be a Totally Bounded Ring And Consider The Folmentioning

confidence: 98%

Extending Ring Topologies

Comfort¹,

Remus²,

Szambien³

2000

Journal of Algebra

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“…In his paper on compact rings [10] (ii) Choose p f e A* such that p'A e Λ^(O). If x e p'A, then there exists y e A such that x = p'y and hence x + p'A -p'y + p'A = p'A.…”

Section: And)mentioning

confidence: 99%

“…In [10,Theorem 5], Warner places the following conditions on a topological ring which has no divisors of zero.…”

Section: ) Given a Nonzero Element P Of R Then P^iφ) Is A Base For mentioning

confidence: 99%

“…Warner has given necessary and sufficient conditions for a topological ring which has no zero divisors to be openly embeddable in a topological division ring [10,Theorem 5]. It is easy to see that the ring (C£, ^~*) does not satisfy these conditions.…”

mentioning

confidence: 99%

“…If the ring satisfies the conditions given by Warner in [10], then the topology which we will define has the property that the quotient field with this topology is a topological field with the ring topologically embedded as an open set. In general, however, the field topology will reflect only part of the algebraic and the topological structure of the ring and will not necessarily be compatible with the field structure.…”

mentioning

confidence: 99%

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