A new notion of semiprime submodules
Masood Aryapoor
TL;DR
The paper extends the notion of semiprimens to submodules of modules over commutative rings and shows that, for finitely generated modules, semiprime submodules coincide with radical submodules (intersections of prime submodules). It provides a constructive framework: a submodule’s radical can be computed as the union of an increasing sequence $N^{(1)},N^{(2)},\dots$ with $N^{(1)}$ generated by elements $m$ satisfying $m\in (N:m)M$, and each $N^{(i)}$ defined by the same rule, yielding $\sqrt{N}=\bigcup_i N^{(i)}$. The work also establishes a quotient-compatibility via a 1-1 correspondence between semiprime submodules containing a fixed submodule and semiprime submodules of the quotient, and situates these results within a broader framework that extends CimpriPrime’s ideas to general modules. Overall, the results provide a concrete, computation-friendly characterization of radical submodules in finitely generated modules and clarify the relationship between semiprimeness and radicality.
Abstract
We introduce a new concept of a semiprime submodule. We show that a submodule of a finitely generated module over a commutative ring is semiprime if and only if it is radical, that is, an intersection of prime submodules. Using our notion, we also provide a new characterization of radical submodules of finitely generated modules over commutative rings.