Let R be an associative ring with identity and let M be a unitary left R–module. As a generalization of small submodule , we introduce Jacobson–small submodule (briefly J–small submodule ) . We state the main properties of J–small submodules and supplying examples and remarks for this concept . Several properties of these submodules are given . Also we introduce Jacobson–hollow modules ( briefly J–hollow ) . We give a characterization of J–hollow modules and gives conditions under which the direct sum of J–hollow modules is J–hollow . We define J–supplemented modules and some types of modules that are related to J–supplemented modules and introduce properties of this types of modules . Also we discuss the relation between them with examples and remarks are needed in our work.
Let R be a ring with identity and M is a unitary left R–module. M is called J–lifting module if for every submodule N of M, there exists a submodule K of N such that M = K ⊕ K ′ , K ′ ⊆ M and N ∩ K ′ ≪ J K ′ . The am of this paper is to introduce properties of J–lifting modules. Especially, we give characterizations of J–lifting modules.We introduce J–coessential submodule as a generalization of coessential submodule . Finally, we give some conditions under which the quotient and direct sum of J–lifting modules is J–lifting.
As input patterns are presented to ANN networks, output patterns are produced. ANN’s neurons constitute layers: the output layer, one or more hidden layers, and the input layer. Hence, information flows to the output layer from the input layer via the hidden layer, with the latter forming a platform for input-output layer association (Meier & Rix, 1994). One of the findings in the study by Nazzal and Tatari (2013), who used ANN for backcalculation of flexible pavement moduli, it was observed that ANN exhibits the capability of predicting layer moduli values of pavements with success; with FWD-enabled field deflection measurements on focus. Similarly, it was noted that ANN adoption yields a significant reduction in computation time while simplifying the backcalculation process.
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