Ideal Containment in Commutative Rings

Let $R$ be a commutative ring with identity and $J$ a nonzero ideal of $R$. Then $J$ is said to be a big ideal if whenever $I\subsetneqq J$J is a subideal of $J$, $I^{n}\subsetneqq J^{n}$for every $n\geq 1$; and $R$ itself is a big ideal ring provided that every non-zero ideal of $R$ is a big ideal. Also $J$ is said to be an upper big ideal if whenever $J\subsetneqq I$, $J^{n}\subsetneqq I^{n}$ for every $n\geq 1$. The notions of big ideals and upper big ideals are completely different, however $R$ is a big ideal ring if and only if every ideal is an upper big ideal. In this project we study the notions of big ideals, upper big ideals and big ideal rings in different contexts of commutative rings such us integrally closed domains, pullbacks constructions and trivial ring extensions.