Duprime and dusemiprime modules

[摘要] A lattice ordered monoid is a structure , where , is a monoid, is a lattice and the binary operation circle plus distributes over finite meets. If M is an element of R-Mod then the set L-M of all hereditary pretorsion classes of sigma [M] is a lattice ordered monoid with binary operation given by alpha :(M) beta := {N is an element of sigma [M] \ there exists A less than or equal to N such that A is an element of alpha and N/A is an element of beta}, whenever alpha, beta is an element of L-M (the subscript in :(M) is omitted if sigma [M] = R-Mod). sigma [M] is said to be duprime (resp. dusemiprime) if M is an element of alpha :(M) beta implies M is an element of alpha or M is an element of beta (resp. M is an element of alpha :(M) -alpha implies M is an element of alpha), for any alpha, beta is an element of L-M. The main results characterize these notions in terms of properties of the subgenerator M. It is shown, for example, that M is duprime (resp. dusemiprime) if M is strongly prime (resp. strongly semiprime). The converse is not true in general, but holds if M is polyform or projective in sigma [M]. The notions duprime and dusemiprime are also investigated in conjunction with finiteness conditions on L-M, such as coatomicity and compactness. (C) 2001 Elsevier Science B.V. All rights reserved.

[发布日期] 2001-12-24