Modular congruences and the Brown-McCoy radical for semigroups

Abstract:

The Brown-McCoy radical ${R_{{G^0}}}$ for semigroups with zero is characterized in terms of modular two-sided congruences. The general notion of the $\mathcal {C}$-radical of a semigroup is used to prove that ${R_{{G^0}}}$ is the ${\rho _s}$-class containing zero, where ${\rho _s}$ is the intersection of all modular maximal two-sided congruences of S. Thus when ${\rho _s}$ is the identity relation, ${R_{{G^0}}} = 0$ and S is isomorphic to a subdirect product of congruence-free semigroups with zero and identity. We also link ${R_{{G^0}}}$ to representation theory.