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In this paper, we extend some results of D. Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.
The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, . We show:
Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m, or α > ω and , then G admits a pseudocompact group topology of weight α.
Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies .
Theorem 5.2(b). If G is divisible Abelian with , then G admits at most -many...
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