Rings in which every semiprime left ideal is an intersection of prime left ideals
Abstract
We study rings over which every semiprime submodule of every module is an intersection of prime submodules. This holds if semiprime two-sided ideals satisfy ACC, if principal two-sided ideals are finitely generated as left ideals, or if the ring is PI. Over Jacobson PI rings, every finitely generated module is Jacobson: its prime submodules are intersections of maximal submodules. For affine PI algebras this extends the Amitsur-Procesi theorem to modules. In every ring with involution, real two-sided ideals are intersections of real prime two-sided ideals. The same holds for real submodules of modules over PI rings and hereditary Noetherian prime rings, but can fail for real left ideals in Noetherian domains. Formally real affine PI algebras need not be real Jacobson: their real prime two-sided ideals need not be intersections of real primitive ideals. We give a sufficient positivity criterion. Over real Jacobson PI rings, finitely generated modules are real Jacobson: their real prime submodules are intersections of real maximal submodules. Over finitely generated free associative algebras, every finitely generated semiprime submodule of a finite free module is a finite intersection of finitely generated prime submodules and an intersection of maximal submodules of finite codimension. Semiprime hulls of finitely generated submodules are finitely generated. Evaluation at finite-dimensional irreducible representations and nonzero vector tuples gives a Hilbert Nullstellensatz over arbitrary fields. For free algebras with involution, the corresponding real results are known over the real or complex numbers when the involution permutes the generators. We extend real prime decomposition to arbitrary fields and involutions, and give new algebraic proofs of the real Nullstellensatz and finite generation of real radicals, without the GNS construction or Groebner bases.
Related
- On a finite group in which every non-supersolvable maximal subgroup is a TI-subgroup
- Extension of valuation rings as limits of complete intersection algebras
- On a finite group in which every maximal subgroup of order divisible by p containing the normalizer of some Sylow subgroup is nilpotent
- On a finite group in which every non-nilpotent maximal invariant subgroup of order divisible by p is normal
- Bricks in which every vertex is incident with a forcing edge
BibTeX
@misc{cimpric2026rings,
title = {{Rings in which every semiprime left ideal is an intersection of prime left ideals}},
author = {Jakob Cimprič},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03967},
eprint = {2611.03967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03967},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rings in which every semiprime left ideal is an intersection of prime left ideals
AU - Jakob Cimprič
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03967
UR - https://arxiv.org/abs/2611.03967
AB - We study rings over which every semiprime submodule of every module is an intersection of prime submodules. This holds if semiprime two-sided ideals satisfy ACC, if principal two-sided ideals are finitely generated as left ideals, or if the ring is PI. Over Jacobson PI rings, every finitely generated module is Jacobson: its prime submodules are intersections of maximal submodules. For affine PI algebras this extends the Amitsur-Procesi theorem to modules. In every ring with involution, real two-sided ideals are intersections of real prime two-sided ideals. The same holds for real submodules of modules over PI rings and hereditary Noetherian prime rings, but can fail for real left ideals in Noetherian domains. Formally real affine PI algebras need not be real Jacobson: their real prime two-sided ideals need not be intersections of real primitive ideals. We give a sufficient positivity criterion. Over real Jacobson PI rings, finitely generated modules are real Jacobson: their real prime submodules are intersections of real maximal submodules. Over finitely generated free associative algebras, every finitely generated semiprime submodule of a finite free module is a finite intersection of finitely generated prime submodules and an intersection of maximal submodules of finite codimension. Semiprime hulls of finitely generated submodules are finitely generated. Evaluation at finite-dimensional irreducible representations and nonzero vector tuples gives a Hilbert Nullstellensatz over arbitrary fields. For free algebras with involution, the corresponding real results are known over the real or complex numbers when the involution permutes the generators. We extend real prime decomposition to arbitrary fields and involutions, and give new algebraic proofs of the real Nullstellensatz and finite generation of real radicals, without the GNS construction or Groebner bases.
ER -