2012/02/02 by John A. Beachy, Mahmood Behboodi, Beachy, John A. +3
Mathematics · #16D50 #16D60 #16N60 #16S38 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1202.0392
openalex publication_date 2012/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a fixed left R-module. For a left R-module X, we introduce the\nnotion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R,\nwhich coincides with prime (resp. semiprime) submodule of X. Other concepts\nencountered in the general theory are M-m-system sets, M-n-system sets, M-prime\nradical and M-Baer's lower nilradical of modules. Relationships between these\nconcepts and basic properties are established. In particular, we identify\ncertain submodules of M, called "prime M-ideals", that play a role analogous to\nthat of prime (two-sided) ideals in the ring R. Using this definition, we show\nthat if M satisfes condition H (defined latter) and HomR(M,X)\≠ 0 for all\nmodules X in the category \σ[M], then there is a one-to-one correspondence\nbetween isomorphism classes of indecomposable M-injective modules in \σ[M]\nand prime M-ideals of M. Also, we investigate the prime M-ideals, M-prime\nsubmodules and M-prime radical of Artinian modules.\n