Logic Seminar

The universal *-regularization of a ring

by Sonia L'Innocente (Università di Camerino)

Aula Riunioni (Dipartimento di Matematica)

Aula Riunioni

Dipartimento di Matematica


Olivier's construction of the universal commutative (von Neumann) regular ring over a commutative ring is generalized to obtain the universal $*$-regular ring over a noncommutative ring $(R, *)$ with involution. The construction of a universal $*$-regular ring proceeds similarly with the Moore–Penrose inverse replacing the role of the group inverse in the construction of universal abelian regular rings.
Under an algebraic point of view, some nice properties of these rings will be described and some nice examples will be given in this context,  such as the Jacobson algebra.
Under the model theoretical point of view, we can observe that
the involution of $(R, *)$ induces an involution on the modular lattice $L(R, 1)$ of positive primitive formulae in the language of left $R$-modules. It is shown that $*$-regular ring coordinatizes the quotient lattice of $L(R, 1)$ modulo the least congruence for which the involution designates an orthogonal complement.