On clean-like conditions in some commutative ring extensions

In this paper, we introduce new classes of rings, namely, \(n^{*}\) $n_{}^{\ast }$ -nil clean ring, \(\sum \) $\sum$ -nil clean ring and BB-ring, and we study the possible transfer of the notions of \(n^{*}\) $n_{}^{\ast }$ -nil clean ring, \(\sum \) $\sum$ -nil clean ring, k-good ring, nil-good ring and BB-ring to various context of commutative ring extensions such as homomorphic image, direct product, power series ring and amalgamation ring, with applications to the transfer of these properties in trivial ring extension. Our work is motivated by an attempt to generate new original classes of rings possessing these properties.