‎In this paper‎, ‎we introduce the notion of the radical of a $PMV$-algebra $A$ and we charactrize radical $A$ via elements of $A$‎. ‎Also‎, ‎we introduce the notion of the radical of a $cdot$-ideal in $PMV$-algebras‎. ‎Several characterizations of this radical is given‎. ‎We define the notion of a semimaximal $cdot$-ideal in a $PMV$-algebra‎. ‎Finally we show that $A/I$ has no nilpotent elemen...

Abstract Krull’s Maximal Ideal Theorem (MIT) is one of the most prominent incarnations Axiom Choice (AC) in ring theory. For many a consequence AC, constructive counterparts are well within reach, provided attention turned to syntactical underpinning problem at hand. This viewpoints revised Hilbert Programme commutative algebra, which will here be carried out for MIT and several related classic...