The Igusa-Todorov $\phi$-dimension on Morita context algebras

TL;DR

This paper investigates the $$-dimension in Morita context algebras, showing finiteness under certain conditions and exploring asymmetry between an algebra and its opposite.

Contribution

It establishes conditions for finite $$-dimension in Morita context algebras and demonstrates the asymmetry of $$-dimension between an algebra and its opposite.

Findings

Morita context algebras with zero bimodule morphisms have finite $$-dimension under certain hypotheses.

The $$-dimension of an algebra can differ from that of its opposite, showing asymmetry.

The $$-dimension is not necessarily symmetric for finite dimensional algebras.

Abstract

In this article we prove that, under certain hypotheses, Morita context algebras that have zero bimodule morphisms have finite $\phi$-dimension. We also study the behaviour of the $\phi$-dimension for an algebra and its opposite. In particular we show that the $\phi$-dimension of an Artin algebra is not symmetric, i.e. there exists a finite dimensional algebra $A$ such that $\phi\dim (A) \not = \phi\dim (A^{op})$.