Maximal Elements and Prime Elements in Lattice Modules


Abstract in English

In this paper we study the relationship between the maximal (prime) elements of M and the maximal (prime) elements of L. We show that, if L is a local lattice and the greatest element of M is weak principal, then M is local . Then we define the Jacobson radical of M and denote it by J(M) and we study its relationship with the Jacobson radical of L (J(L)) . Afterwards, we define the semiprime element in a lattice module M, and we show that the definitions of prime element and semiprime element are equivalent when the greatest element of M is multiplication and we study the properties equivalent to the properties of prime element in lattice module .

References used

DILWORTH, R. P. 1962. Abstract commutative ideal theory, pacific J. Math ., 12 , 481 – 498
JOHNSON, J. A. 1970. a–adic completions of Noetherian lattice modules. Fund. Math., 66, 341 – 371
JOHNSON, E. W. and JOHNSON, J. A. 1970. Lattice modules over semi–local Noether lattices. Fund. Math., 68, 187–201

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