Semiprime and real submodules of arbitrary modules are intersections of prime and real prime submodules
Abstract
Let R be a commutative ring. A submodule N of an R-module M is semiprime if m∈(N:m)M implies m∈ N. It is known that every semiprime submodule of Rn is an intersection of prime submodules, and Aryapoor extended this to finitely generated modules. We show that no finiteness assumption is needed: every semiprime submodule of an \emph{arbitrary} R-module is the intersection of the prime submodules containing it. The proof is short and uses neither coordinates nor presentations; its key step is that a nonzero element q of a semiprime module Q satisfies sq∉p Q for all s∉p, where p is any minimal prime over the annihilator of q. We define an R-module M to be real if ∑i mi⊗ mi=0 in M⊗RM implies mi=0 for all i, and a submodule N⊆ M real if the module M/N is real. We prove the corresponding result for real submodules: every real submodule of an arbitrary module is the intersection of the real prime submodules containing it. On Rn our notion of realness coincides with the coordinate definition used in earlier work on real Nullstellensätze for free modules.
Related
- Euclidean modules, submodules of their direct sums and strong reduced echelon form
- Breaker Chains and Splitting of Interval Submodules in Zigzag Modules
- Rings in which every semiprime left ideal is an intersection of prime left ideals
- Regularity of Symbolic Powers of Polymatroidal Ideals and Supermodular Prime Intersections
- Periodic Tilings of Cardinality Twice a Prime or Nine in Arbitrary Dimension
BibTeX
@misc{cimpric2026semiprime,
title = {{Semiprime and real submodules of arbitrary modules are intersections of prime and real prime submodules}},
author = {Jakob Cimprič},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15869},
eprint = {2610.15869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15869},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Semiprime and real submodules of arbitrary modules are intersections of prime and real prime submodules
AU - Jakob Cimprič
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15869
UR - https://arxiv.org/abs/2610.15869
AB - Let R be a commutative ring. A submodule N of an R-module M is semiprime if m∈(N:m)M implies m∈ N. It is known that every semiprime submodule of R^n is an intersection of prime submodules, and Aryapoor extended this to finitely generated modules. We show that no finiteness assumption is needed: every semiprime submodule of an \emph{arbitrary} R-module is the intersection of the prime submodules containing it. The proof is short and uses neither coordinates nor presentations; its key step is that a nonzero element q of a semiprime module Q satisfies sq∉p Q for all s∉p, where p is any minimal prime over the annihilator of q. We define an R-module M to be real if ∑_i m_i⊗ m_i=0 in M⊗_RM implies m_i=0 for all i, and a submodule N⊆ M real if the module M/N is real. We prove the corresponding result for real submodules: every real submodule of an arbitrary module is the intersection of the real prime submodules containing it. On R^n our notion of realness coincides with the coordinate definition used in earlier work on real Nullstellensätze for free modules.
ER -