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Arithmetically Related Ideal Topologies and the Infinitude of Primes
Abstract
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, 373-391
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, 373-391
