Let S be the left bialgebroid End BAB over the centralizer R of a right D2 algebra extension A | B, which is to say that its tensor-square is isomorphic as A-B-bimodules to a direct summand of a finite direct sum of A with itself. Without an antipode, we prove that the left endomorphism algebra is a left S-Galois extension of A, and find a formula for the inverse Galois mapping. As a corollary,...

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 excha...

A module is called uniseriat if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings, proved [21, Theorem 171 that every finitely generated module over a serial ring is a direct sum of uniserial modules. In section one we g...

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel described the structure of isogeny graphs for elliptic curves and showed that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth first search algorithm in the graph. In dimension 2,...

As mentioned several times in class, the arithmetic of quaternionic Shimura varieties is strongly controlled by the behavior of the class of CM points (just as in the case of modular curves). Just as for elliptic curves, if o is an order in an imaginary quadratic field K, one has a notion of a QM-surface with o-CM. Namely, let ι : O ↪→ A be a QM structure on an abelian surface (we are still wor...

We recall that a ring R is called near pseudo-valuation ring if every minimal prime ideal is a strongly prime ideal. Let R be a commutative ring, σ an automorphism of R and δ a σderivation of R. We recall that a prime ideal P of R is δ-divided if it is comparable (under inclusion) to every σ-invariant and δ-invariant ideal I (i.e. σ(I) ⊆ I and δ(I) ⊆ I) of R. A ring R is called a δ-divided ring...

Let k be a field, G an abelian group with a bicharacter, A a colour algebra; i.e., an associative graded k-algebra with identity, C a graded A-coring that is projective as a right A-module, C∗ the graded dual ring of C and Λ a left graded C-comodule that is finitely generated as a graded right C∗-module. We give necessary and sufficient conditions for projectivity and flatness of a graded modul...

A ring R is called an E-ring if the canonical homomorphism from R to the endomorphism ring End (RZ) of the additive group RZ, taking any r ∈ R to the endomorphism left multiplication by r turns out to be an isomorphism of rings. In this case RZ is called an E-group. Obvious examples of E-rings are subrings of Q. However there is a proper class of examples constructed recently, see [8]. E-rings ...

We answer in the negative a question of Hartley about representations finite groups, by constructing examples simple groups with arbitrarily large whose endomorphism ring consists just scalars. show as consequence that there are automorphisms locally group SL(∞,Fq) trivial centralizer. The smallest our is A6 q=9.

A ring $R$ is called left comorphic if for every $a\in R$ there exists $b\in
 such that the and right annihilators satisfy $Ra=l(b)$ and
 $r(a)=bR$. In this paper, Abelian groups with comorphic
 endomorphism rings are completely determined.