QUASI-RADICALS AND RADICALS IN CATEGORIES

A. Buys and S. Veldsman

Abstract: In a category $\Cal K$, if $\Cal E$ is a class of epimorphisms and $\Cal M$ a class of monomorphisms, a funtion $J_r$ called an $(\Cal E, \overline{\Cal M})$-quasi-radical, is defined which assigns to an object an $\Cal M$-sink and a function $J_c$, called an $(\overline{\Cal E},\Cal M)$-quasi-coradical, is defined which assigns to an object an $\Cal E$-source. With $J_r$ are associated two object classes {\bf R}$_r$ and {\bf S}$_r$ called the quasi-radical class and the quasi-semisimple class respectively. With $J_c$ are associated two object classes {\bf R}$_c$ and {\bf S}$_c$, called the quasi-coradical class and the quasi-cosemisimple class respectively. Using these notions, an $(\Cal E,\overline{\Cal M})$-radical is a pair $(J_r,J_c)$ where $J_r$, is a quasiradical, $J_c$ a quasi-coradical and for which ${\bold R}_r={\bold R}_c$ and ${\bold S}_r={\bold S}_c$. Among others it is shown that ${\bold R}_r={\bold R}_c$ is a radical class and ${\bold S}_r= {\bold S}_c$ is a semisimple class.

Classification (MSC2000): 18E40

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