Pace P. Nielsen

Odd perfect numbers have at least nine distinct prime factors

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

NILPOTENT IDEALS IN POLYNOMIAL AND POWER SERIES RINGS

Given a ring R and polynomials f(x), g(x) ∈ R[x] satisfying f(x)Rg(x) = 0, we prove that the ideal generated by products of the coefficients of f(x) and g(x) is nilpotent. This result is generalized, and many well known facts, along with new ones, concerning nilpotent polynomials and power series are obtained. We also classify which of the standard nilpotence properties on ideals pass to polynomial rings or from ideals in polynomial rings to ideals of coefficients in base rings. In particular, we prove that if I ≤ R[x] is a left T-nilpotent ideal, then the ideal formed by the coefficients of polynomials in I is also left T-nilpotent.

Countable Exchange and Full Exchange Rings

We show that a Dedekind-finite, semi-π-regular ring with a “nice” topology is an ℵ0-exchange ring, and the same holds true for a strongly clean ring with a “nice” topology. We generalize the argument to show that a Dedekind-finite, semi-regular ring with a “nice” topology is a full exchange ring. Putting these results in the language of modules, we show that a cohopfian module with finite exchange has countable exchange, and all modules with Dedekind-finite, semi-regular endomorphism rings are full exchange modules. These results are generalized further.

ABELIAN EXCHANGE MODULES

ABSTRACT Let Mk be a right k-module with endomorphism ring E=End(Mk). We prove that if E is an Abelian exchange ring, then Mk has the full exchange property. We also give an extension of this result in the case E is regular.

Idempotent lifting and ring extensions

We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring R for which idempotents lift modulo the Jacobson radical J(R), but idempotents do not lift modulo J(𝕄2(R)). Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if I and J are ideals of R for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over I + J. On the positive side, if I and J are enabling ideals in R, then I + J is also an enabling ideal. We show that if I ⊴ R is (weakly) enabling in R, then I[t] is not necessarily (weakly) enabling in R[t] while I⟦t⟧ is (weakly) enabling in R⟦t⟧. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.

Odd perfect numbers, Diophantine equations, and upper bounds

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 1012P 2N 10 and N has at least 101 prime factors (counting multiplicity). If k is the number of distinct prime factors, then as proved in [12, 13] we have k ≥ 9 and N 10. Starting in §2, readers should be familiar with basic facts about odd perfect numbers, including knowledge of congruence restrictions related to the special prime. As this paper is an extension of the methods used in [13], starting in §3 the reader should be familiar with the ideas in that paper. 1. A better upper bound Let N be a positive integer. Following the literature, N is said (in increasing order of generality) to be perfect when σ(N)/N = 2, multiperfect when σ(N)/N ∈ Z, and n/d-perfect when σ(N)/N = n/d. For simplicity, we will always assume n, d ∈ Z>0. Note that n/d does not need to be in lowest terms. Writing N = ∏k i=1 p ei i where p1 < . . . < pk are the prime divisors of N , the equation σ(N)/N = n/d 2010 Mathematics Subject Classification. Primary 11N25, Secondary 11Y50.

Derivations and bounded nilpotence index

We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x;δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about locally nilpotent PI rings.

Sums of units in self-injective rings

We prove that if no field of order less than n + 2 is a homomorphic image of a right self-injective ring R, then for any element a ∈ R and central units u1, u2, …, un ∈ U(R) there exists a unit u ∈ U(R) such that a + uiu ∈ U(R) for each i ≥ 1.

Simplifying Smoktunowicz's Extraordinary Example

At the turn of the 21st century Agata Smoktunowicz constructed the first example of a nil algebra over a countable field such that the polynomial ring over the algebra is not nil. This answered an old question of Amitsur. We present a simplification of the example.

Sieve methods for odd perfect numbers

Using a new factor chain argument, we show that 5 does not divide an odd perfect number indivisible by a sixth power. Applying sieve techniques, we also find an upper bound on the smallest prime divisor. Putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 10^8 < p < 10^1000. These results are generalized to much broader situations.