We prove that all reversible rings are McCoy, generalizing the fact that both commutative and reduced rings are McCoy. We then give an example of a semi-commutative ring that is not right McCoy. At the same time, we also show that semi-commutative rings do have a property close to the McCoy condition.  2005 Elsevier Inc. All rights reserved.

Abstract. An odd perfect number, N , is shown to have at least nine distinct prime factors. If 3 N then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.

We investigate relations between the McCoy property and other standard ring theoretic properties. For example, we prove that the McCoy property does not pass to power series rings. We also classify how the McCoy property behaves under direct products and direct sums. We prove that McCoy rings with 1 are Dedekind finite, but not necessarily Abelian. In the other direction, we prove that duo rings, and many semi-commutative rings, are McCoy. Degree variations are defined, studied, and classified. The McCoy property is shown to behave poorly with respect to Morita equivalence and (infinite) matrix constructions.

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We obtain a new upper bound for odd multiperfect numbers. If N is an odd perfect number with k distinct prime divisors and P is its largest prime divisor, we find as a corollary that 10 12 P 2 N < 2 4 k . Using this new bound, and extensive computations, we derive the inequality k ≥ 10.