2021/02/09 by Jaka Cimprič, Cimprič, Jaka
Mathematics · #13C10 #14A22 #16D25 #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2102.04786
openalex publication_date 2021/02/09 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Let R be a commutative ring with 1 and n a natural number. We say that\na submodule N of Rn is semiprime if for every f=(f1,\…,fn) \∈ Rn\nsuch that fi f \∈ N for i=1,\…,n we have f \∈ N. Our main result\nis that every semiprime submodule of Rn is equal to the intersection of all\nprime submodules containing it. It follows that every semiprime left ideal of\nMn(R) is equal to the intersection of all prime left ideals that contain it.\nFor R=k[x1,\…,xd] where k is an algebraically closed field we can\nrephrase this result as a Nullstellensatz for Mn(R): For every\nG1,\…,Gm,F \∈ Mn(R), F belongs to the smallest semiprime left ideal\nof Mn(R) that contains G1,\…,Gm iff for every a \∈ kd and v \∈\nkn such that G1(a)v=\…=Gm(a)v=0 we have F(a)v=0.\n