چکیده:
Abstract. Let R be a commutative ring with identity and let M be a unital R-module. In this
paper we study the various properties of prime submodules. Also we give a new equivalent
conditions for a minimal prime submodules of an R-module to be a finite set, whenever R is
a Noetherian ring. Finally we prove the Prime avoidance Theorem for modules in different
states.
خلاصه ماشینی:
"In section 4 we prove some new results about the finiteness of the set of minimal prime submodules of an R-module.
9. Let be a Noetherian ring and be a finitely generated R-module and be a -prime submodule of .
9. Let be a Noetherian ring and be a finitely generated R-module and be a -prime submodule of .
4. Minimal prime submodules The following lemma is needed in the proof of the first main result of this section.
In the next theorem we present a new conditions that an R- module M has only a finite number of minimal prime submodules, whenever R is a Noetherian ring, which is a generalization of [2, Theorem 2.
5. Let R be a Noetherian ring, M an R-module and B be a proper submodule of M such that any minimal prime submodule over B is finitely generated.
Before bringing the next definition, recall that for any ideal I of a Noetherian ring, the arithmetic rank of I, denoted by ara(I), is the least number of elements of I required to generate an ideal which has the same radical as I, i.
8. Let R be a Noetherian ring, / a finitely generated R-module and N be a proper submodule of M.
8. Let R be a Noetherian ring, / a finitely generated R-module and N be a proper submodule of M.
8. Let R be a Noetherian ring, / a finitely generated R-module and N be a proper submodule of M."