let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

In this paper, we investigate the ideal theory in BCK-algebras and we define the notions of primary and P -primary ideals. Then we show that in bounded implicative BCK-algebras, if an ideal has a primary decomposition, then it has a reduced primary decomposition. In the follow, we extend the above concepts to X-modules (Extended BCK-modules) and we introduce the notions of primary and P -primar...

The first part is about primary decomposition. After reviewing the basic definitions, we survey the compatibility, independence, and linear growth properties that have been known. Then, we prove the linear growth property of primary decomposition for a new family of modules. In the remaining sections, we study secondary representation, which can be viewed as a dual of primary decomposition. Cor...

We study the following properties about primary decomposition over a Noetherian ring R: (1) For finitely generated modules N ⊆ M and a given subset X = {P1, P2, . . . , Pr} ⊆ Ass(M/N), we define an X-primary component of N ( M to be an intersection Q1 ∩ Q2 ∩ · · · ∩ Qr for some Pi-primary components Qi of N ⊆ M and we study the maximal X-primary components of N ⊆ M ; (2) We give a proof of the ...