There are two different notions for symbolic powers of ideals existing in the literature, one defined terms associated primes, other minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, Hoa, and Trung, we interpret both as suitable saturations ordinary powers. We prove a binomial expansion formula saturated sums ideals. This gives uniform treatment array new results such sums...

In this paper, by using elementary tools of commutative algebra,we prove the persistence property for two especial classes of rings. In fact, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1,...

For a radical ideal I, the symbolic power I is the collection of elements that vanish to order at least p at each point of Zeros(I). If I is actually prime, then I is the I-associated primary component of I; if I is only radical, writing I = C1 ∩ · · · ∩Cs as an intersection of prime ideals, I = C (p) 1 ∩ · · · ∩ C (p) s . The inclusion I ⊆ I always holds, but the reverse inclusion holds only i...

Abstract Fix a poset Q on $\{x_1,\ldots ,x_n\}$ . A -Borel monomial ideal $I \subseteq \mathbb {K}[x_1,\ldots ,x_n]$ is whose monomials are closed under the Borel-like moves induced by I principal ideal, denoted $I=Q(m)$ , if there m such that all minimal generators of can be obtained via from In this paper we study powers ideals. Among our results, show $Q(m)$ agree with their symbolic powers,...