Radicals of semirings

Abstract

It is shown that the classes [Formula: see text] and [Formula: see text] of semirings are radical classes, where [Formula: see text] is the class of subtractive-simple right [Formula: see text]-semimodules and [Formula: see text] is the class of right [Formula: see text]-semimodules isomorphic to [Formula: see text] for some maximal-subtractive right ideal [Formula: see text] of [Formula: see text]. We define the lower Jacobson Bourne radical [Formula: see text] and upper Jacobson Bourne radical [Formula: see text] of [Formula: see text]. For a semiring [Formula: see text], [Formula: see text] holds, where [Formula: see text] is the Jacobson Bourne radical of [Formula: see text]. The radical [Formula: see text] and also coincides with [Formula: see text], if we restrict the class [Formula: see text] to additively cancellative semimodules[Formula: see text] The upper radical [Formula: see text] and [Formula: see text][Formula: see text], if [Formula: see text] is additively cancellative. Further, [Formula: see text], if [Formula: see text] is a commutative semiring with [Formula: see text] The subtractive-primitiveness and subtractive-semiprimitiveness of [Formula: see text] are closely related to the upper radical [Formula: see text] Finally, we show that [Formula: see text]-semisimplicity of semirings are Morita invariant property with some restrictions.