Associative Dialgebras from a structural viewpoint

TL;DR

This paper investigates the structure of associative dialgebras, revealing that interesting cases occur when the algebra is not semiprime, and explores specific classes like zero-cubed and two-dimensional dialgebras.

Contribution

It characterizes associative dialgebras based on their structural properties and identifies conditions under which they derive from associative algebras with combined products.

Findings

Interesting dialgebras are non-semiprime

Associative dialgebras with certain properties originate from associative algebras with identified products

Classification of zero-cubed and two-dimensional associative dialgebras

Abstract

In this note we study associative dialgebras proving that the most interesting such structures arise precisely when the algebra is not semiprime. In fact the presence of some "perfection" property (simpleness, primitiveness, primeness or semiprimeness) imply that the dialgebra comes from an associative algebra with both products identified. We also describe the class of zero-cubed algebras and apply its study to that of dialgebras. Finally we describe two-dimensional associative dialgebras.