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.
