ON THE RADICAL OF IDEALS IN SEMIRINGS

. The radical of a subset A of a semiring S is the set of elements whose exponents are preserved in the same set A. In commutative semirings with unity, it was known that the radical of a two-sided ideal is described by the intersection of all prime ideals and the radical of such a two-sided ideal is still a two-sided ideal. This known result holds if the commutativity and the existence of unity are provided. In the present paper, we consider the behavior of the radicals of various ideals in semirings, not only the radicals of two-sided ideals. Moreover, we skip such two conditions: commutativity and the existence of unity. The interesting point we provide by an example is that the radical of an ideal may not be an ideal. Therefore, by using the properties of such ideals in semirings, we consider the conditions that preserve the ideal properties upon radical formation. Then, we conclude all conditions that the radical of ideals in semirings preserves ideals properties.