E-ideals in exponential polynomial ring

Abstract: We investigate exponential ideals within the context of exponential polynomial rings over exponential fields. We establish two distinct notions of maximality for exponential ideals and explore their relationship to primeness. These three concepts--prime, maximal, and E-maximal--are shown to be independent, in contrast to the classical scenario. Furthermore, we demonstrate that, over an algebraically closed field K, the correspondence between points of $K^n$ and maximal exponential ideals of the ring of exponential polynomials breaks down. Finally, we introduce and characterize exponential radical ideals. We investigate exponential ideals in the exponential polynomial ring over an exponential field. We study two notions of maximality for exponential ideals, and relate them to primeness. These three notions are independent, unlike in the classical case. We also show that over an algebraically closed field K the correspondence between points of K^n and maximal ideals of the ring of exponential polynomials does not hold.