Bounded topologies on endomorphism rings

We prove in this note that the ring of endomorphisms of an infinite bounded Abelian group admits a nondiscrete right bounded ring topology. We give an example of an Abelian group whose ring of endomorphisms admits both nondiscrete left and right bounded topologies but does not admit a nondiscrete bounded ring topology.

SQUARE-FREE DISCRIMINANTS OF FROBENIUS RINGS

Let E be an elliptic curve over ℚ. We know that the ring of endomorphisms of its reduction modulo an ordinary prime p is an order of the quadratic imaginary field generated by the Frobenius element πp. However, except in the trivial case of complex multiplication, very little is known about the fields that appear as algebras of endomorphisms when p varies. In this paper, we study the endomorphism ring by looking at the arithmetic of [Formula: see text], the discriminant of the characteristic polynomial of πp. In particular, we give a precise asymptotic for the function counting the number of primes p up to x such that [Formula: see text] is square-free and in certain congruence class fixed a priori, when averaging over elliptic curves defined over the rationals. We discuss the relation of this result with the Lang–Trotter conjecture, and some other questions on the curve modulo p.

Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms

We study the interplay between canonical heights and endomorphisms of an abelian variety A over a number field k. In particular we show that whenever the ring of endomorphisms defined over k is strictly larger than there will be -linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of A(k).

Full subrings of E-rings

A ring R is said to be an E-ring if the map R → of E (R)+ into the ring of endomorphisms of its additive group via a ↪ al = left multiplication by a, is an isomorphism. In this note torsion free rings R for which the group Rl, of left multiplication maps by elements of R, is a full subgroup of E(R)+ will be considered. These rings are called TE-rings. It will be shown that TE-rings satisfy many properties of E-rings, and that unital TE-rings are E-rings. If R is a TE-ring, then E(R+) is an E-ring, and E(R+)+ / is bounded. Some results concerning additive groups of TE-rings will be obtained.

Related representation theorems for rings, semi-rings, near-rings and semi-near-rings by partial transformations and partial endomorphisms

Fundamental statements for (associative) rings are that (a) the endomorphisms of each commutative group (U, +) form a ring and (b) eachring may be embedded in such a ring of endomorphisms. In order to generalise these theorems to groups and rings whose addition may not be commutative, one has to deal with partial endomorphisms. But thesering-theoretical Theorems 4a and 4b turn out to be specialisations of similarones for semi-near-rings, near-rings and semirings, developed here inSection 2 after some preliminaries on semi-near-rings in Section 1. A glance at Definition 1 and the ring-theoretical theorems and remarks at the end of Section 2 may give more orientation.

Orthodox bands of modules

In this paper we shall consider orthodox bands of commutative groups, together with a ring of endomorphisms. We shall generalize the concept of a left module by introducing orthodox bands of left modules; we shall also deal with linear mappings, the transpose of a linear mapping and with the dual of an orthodox band of left modules.