In a recent paper, the first author introduced a general theory of corner rings in noncommutative rings that generalized the classical theory of Peirce decompositions. This theory is applied here to the study of the stable range of rings upon descent to corner rings. A ring is called quasi-duo if every maximal 1-sided ideal is 2-sided. Various new characterizations are obtained for such rings. ...

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show these counts are interrelated via partial sums. Moreover, we interpret the study in language graded polynomial rings apply this to mod <mml:se...

Let R be an associative ring. A map σ : R → R is called a ring endomorphism if σ(x+y) = σ(x)+σ(y) and σ(xy) = σ(x)σ(y) for all elements a,b ∈ R. A ring R is said to be rigid if it has only the trivial ring endomorphisms, that is, identity idR and zero 0R . Rigid left Artinian rings were described by Maxson [9] and McLean [11]. Friger [4, 6] has constructed an example of a noncommutative rigid r...

A ring of rank n is a commutative ring that is free of rank n as a Z-module, i.e., isomorphic to Z n as an abelian group. The prototypical example of a ring of rank n is an order in a degree n number field. In this series of lectures, we investigate how rings of small rank n can be explicitly parametrized. The cases n = 4 and n = 5 are particularly interesting, as they are connected to the arit...

Stable isotopes in tree rings could provide palaeoclimate reconstructions with perfect annual resolution and statistically defined confidence limits. Recent advances make the approach viable for non-specialist laboratories. The relevant literature is, however, spread across several disciplines, with common problems approached in different ways. Here we provide the first overview of isotope dend...

LetK/Qp be a finite extension and GK = Gal(K̄/K) the Galois group of an algebraic closure K̄. Let F be a finite field of characteristic p, and VF a finite dimensional F-vector space equipped with a continuous action of GK . The study of the deformation theory of Galois representations was initiated by Mazur [Ma], who showed that if VF has no non-trivial endomorphisms, then it admits a universal d...