The nearly Gorenstein property for numerical duplications and semitrivial extensions

In this thesis we present and study the ideal duplication, a new construction within the class of the relative ideals of a numerical semigroup $S$, that, under specific assumptions, produces a relative ideal of the numerical duplication $SJoin^b E$, for some ideal $E$ of $S$. We prove that every relative ideal of the numerical duplication can be uniquely written as the ideal duplication of two relative ideals of $S$; this allows us to better understand how the basic operations of the class of the relative ideals of $SJoin^b E$ work. In particular, we characterize the ideals $E$ such that $SJoin^b E$ is nearly Gorenstein. With the aim to generalize this construction to commutative rings with unity, we introduce the semitrivial ideal extension, a construction that, starting with an ideal of a commutative ring $R$ with unity and a submodule of a module $M$ over $R$, under specific assumptions, produces an ideal of the semitrivial extension $Rltimes_{phi}M$. Using this tool we characterize a certain family of prime ideals of the semitrivial extension and we completely describe the family of the maximal ideals. Similarly as it was done for the numerical duplication, using the semitrivial ideal extension, we characterize the modules $M$ such that $Rltimes_{phi}M$ is nearly Gorenstein.