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Topologies related to arithmetical properties of integral domains. (English) Zbl 0883.11043
The authors describe several topologies on integral domains and show that some arithmetical properties of the domains, such as the infinitude of primes, can be interpreted in terms of conditions on the topology involved. The point of departure is a paper by S. W. Golomb [Proc. Symp. Prague 1961, 179-186 (1962; Zbl 0116.03404)], where the topology on the set of positive integers was taken to have the primitive arithmetical progressions as a basis for the open sets.
The authors introduce some coset topologies where a basis for the open sets consists of all cosets of all proper ideals of a ring or of all invertible cosets. Under these topologies the multiplicative semigroup of the ring becomes a topological semigroup. The quotient topologies on the semigroup of classes of associate elements of the domain are also studied. A sufficient condition for the infinitude of the set of maximal ideals of the domain is the non-openness of the set of units of the ring. This is the starting point of a deeper study of topological conditions for the existence of infinitely many prime elements of the ring in its cosets. One new result obtained in this way asserts that given any positive integer \(s\), every primitive arithmetical progression contains infinitely many products of \(s\) pseudoprimes.

11N80 Generalized primes and integers
11N25 Distribution of integers with specified multiplicative constraints
11A25 Arithmetic functions; related numbers; inversion formulas
22A99 Topological and differentiable algebraic systems
22A15 Structure of topological semigroups