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Let M be a fixed left R-module. For a left R-module X, weintroduce the notion of M-prime (resp. M-semiprime) submodule of Xsuch that in the case M = R, it coincides with prime (resp. semiprime)submodule of X. Other concepts encountered in the general theory areM-m-system sets, M-n-system sets, M-prime radical and M-Baer’s lowernilradical of modules. Relationships between these concepts and basicproperties are established. In particular, we identify certain submodulesof M, called “primeM-ideals”, that play a role analogous to that of prime(two-sided) ideals in the ring R. Using this definition, we show that if Msatisfies condition H (defined later) and HomR(M,X) ≠ 0 for all mod-ules X in the category σ[M], then there is a one-to-one correspondencebetween isomorphism classes of indecomposable M-injective modules inσ[M] and prime M-ideals of M. Also, we investigate the prime M-ideals,M-prime submodules and M-prime radical of Artinian modules.
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참고문헌 (21)
참고문헌 신청
C. P. Lu / 1984 / Prime submodules of modules / Comment. Math. Univ. St. Paul 33 (1) : 61 ~ 69
G. Krause / 1972 / On fully left bounded left Noetherian ring / J. Algebra 23 : 88 ~ 99
G. Michler / 1972 / Prime right ideals and right Noetherian rings, Proc. Symposium on Theory of Rings, In Ring theory / Academic Press : 251 ~ 255
H. I. Karakas / 1972 / On Noetherian module / METU J. Pure Appl. Sci 5 (2) : 165 ~ 168
J. A. Beachy / 2002 / M-injective modules and prime M-ideals / Comm. Algebra 30 (10) : 4649 ~ 4676
G. Krause / 1972 / On fully left bounded left Noetherian ring / J. Algebra 23 : 88 ~ 99
G. Michler / 1972 / Prime right ideals and right Noetherian rings, Proc. Symposium on Theory of Rings, In Ring theory / Academic Press : 251 ~ 255
H. I. Karakas / 1972 / On Noetherian module / METU J. Pure Appl. Sci 5 (2) : 165 ~ 168
J. A. Beachy / 2002 / M-injective modules and prime M-ideals / Comm. Algebra 30 (10) : 4649 ~ 4676
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