Arithmetically Related Ideal Topologies and the Infinitude of Primes

The late J. Knopfmacher and the author [12] have studied some ties between arithmetic properties of the multiplicative structure of commutative rings with identity and the topologies induced by some coset classes. In the present communication it is shown that the ideas used there are capable of a further extension. Namely, replacing the ideal structure of commutative rings by generalized ideal systems, the so called x-ideals, conditions implying the existence of infinitely many prime x-ideals are found using topologies induced by cosets of x-ideals. This leads to new variants of Fürstenberg topological proof of the infinitude of prime numbers not depending on the additive structure of the underlying integers or commutative rings with identity. as a byproduct we give new proofs of the infinitude of primes based on tools taken from commutative algebra. Mathematics Subject Classification (1991): 11N80, 11N25, 11A41, 11T99, 13A15, 20M25 Keywords: x-ideal, topological semigroup, ideal topology, infinitude of primes, generalized primes and integers, distribution, integers, specified multiplicative constraints, primes, ideals, multiplicative ideal theory, semigroup rings, multiplicative semigroups of rings, multiplicative arithmetical semigroup, semigroup, John Knopfmacher, zeta function, abscissa, convergence, commutative, rings, ring, identities, algebra Quaestiones Mathematicae 24(3) 2001, 373-39