Doctoral defence: Kristo Väljako "On the Morita equivalence of idempotent rings and monomorphisms of firm bimodules“

On 26 August at 11:00 Kristo Väljako will defend his doctoral thesis “On the Morita equivalence of idempotent rings and monomorphisms of firm bimodules” for obtaining the degree of Doctor of Philosophy (in Mathematics).

Supervisor: 

Professor Valdis Laan, University of Tartu

Opponents: 

Professor Mark Lawson, Heriot-Watt University, The United Kingdom

Professor Laiachi El Kaoutit, The University of Granada, Spain

Summary 

The notion of a Morita equivalence for rings with identity first arose in 1958 from the seminal paper by Kiiti Morita. He described when the module categories of two rings with identity are equivalent. Later this situation became known as Morita equivalence of the underlying rings. The resulting Morita theory has proven to be very useful in the development of the theory of rings with identity.

In my thesis I study the Morita equivalence of idempotent rings using various algebraic constructions. A ring is called idempotent if its every element can be expressed as a sum of products of some elements. Idempotent rings are a generalization of rings with identity. In particular, I study the construction of Rees matrix rings and tensor product rings over Idempotent rings. I prove that an idempotent Rees matrix ring is Morita equivalent to its ground ring and that every pseudo-surjectively defined tensor product ring over an idempotent ring is Morita equivalent to that ring. I also introduce and study the enlargements of idempotent rings. In particular, I prove that two idempotent rings are Morita equivalent if and only if they have a joint enlargement. Then I study unitary ideals of Morita equivalent idempotent rings. I will show that the set of unitary ideals of a given ring forms a quantale. I prove that the quantales of unitary ideals of two Morita equivalent idempotent rings are isomorphic. Additionally, the category of firm bimodules over two idempotent rings and especially monomorphisms in this category will be studied. I explicitly show that the categories of firm and closed bimodules over two idempotent rings are equivalent. Finally I will give a description of monomorphisms in the category of firm bimodules over idempotent rings and use that description to show that the lattice of unitary sub-bimodules of a given firm bimodule is isomorphic to the lattice of categorical subobjects of this bimodule.