The depth of a subalgebra B in an algebra A is a number computed by considering tensor powers of modules. The notion touches on various areas of algebra with some nice classification arguments. There are a few different approaches to working with depth: this talk will reveal them.

A left R-module M is called morphic if M/im(f) is isomorphic to ker(f), for every endomorphism f of M, that is if the dual of the Nöther isomorphism theorem holds.
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In this talk I will present Galois correspondence between subalgebras of a Hopf Galois extension and quotients of the structure Hopf algebra. I will also show some characterisations of its closed elements. I will also show some examples and a computation of lattices of generalised quotients of...

Ore localisation is a classical way to obtain ring epimorphisms with a flatness condition. Universal localisations of rings are generalisations of Ore localisations that, although not in general flat, still have interesting homological properties. In this talk we will discuss the notion of...

The ring of Hurwitz integers, being both a left and a right PID, could be thought of as being arithmetically fairly simple. However, the fact that it is not commutative entails some complications, but also some surprises, as well as some interesting open problems. In this talk we will describe...

We introduce a new notion of exact sequences in arbitrary pointed (non-exact) categories. We apply this notion to provide restricted versions of the Short Five Lemma and the Snake Lemma in the category of cancellative right semimodules over a semirings.

Proper classes were introduced by Buchsbaum to axiomatize conditions under which a class of short exact sequences of modules can be computed as Ext groups corresponding to a certain relative homology. In this talk, we shall present some typical ways of generating a proper class from a given class...

We prove a generalized version of a conjecture of A. Tyszka on the relative magnitude of solutions of certain linear systems with integer coefficients. The proof uses combinatorial and linear algebra techniques.

Last year, Izakhian and Rhodes developed a theory of representation of matroids by boolean matrices where all matroids become representable, unlike the case of classical representations over fields. In a joint work with Rhodes, we explore a similar approach in the context of finite graphs,...

A ring R is called right McCoy if whenever non-zero polynomials f (x) and g(x) in R[x] satisfy f (x)g(x) = 0, then f(x)r = 0 for some non-zero r ∈ R. A ring R is Armendariz if f(x)g(x) = 0 implies that all pairwise products of coefficients of f(x) and g(x) are zero. In my talk I am going to...