We introduce the concept of S -primary decomposition for a commutative ring and show that if a ring admits an S -primary decomposition, then it admits a minimal S -primary decomposition. We establish the S -version of Krull’s intersection theorem by using the existence of S -primary decomposition in S -Noetherian domains. As the main result, we prove the Lasker–Noether theorem for nonnil- S -Noetherian rings, thereby extending the classical Lasker–Noether theorem.
Singh et al. (Thu,) studied this question.