Narrow Results

For bounded lattices, we introduce certain Galois connections, called (cyclically) essential, retractable and UC Galois connections, which behave well with respect to concepts of module-theoretic nature involving essentiality. We show that essential retractable Galois connections preserve uniform dimension, whereas essential retractable UC Galois connections induce a bijective correspondence between sets of closed elements. Our results are applied to suitable Galois connections between submodule lattices. Cyclically essential Galois connections unify semi-projective and semi-injective modules, while retractable Galois connections unify retractable and coretractable modules.

We give a relativization of the notions of essential, uniform, closed, injective and extending modules with respect to a left exact preradical on the ring. We also give useful examples and show that several classical properties hold for these kinds of modules.

Recall that an n-by-n generalized matrix ring R is defined in terms of sets of rings (R_i, R_j)- bimodules {M_ij}, and bimodule homomorphisms θ_ijk : M_ij ⊗_RjM_jk → M_ik, where the set of diagonal matrix units {E_ii} form a complete set of orthogonal idempotents. Moreover, an arbitrary ring R with a complete set of orthogonal idempotents has a Peirce decomposition which can be arranged into an n-by-n generalized matrix ring R^π which is isomorphic to R. In this paper, we focus on the subclass T_n of n-by-n generalized matrix rings with θ_iji = 0 for i ≠ j. T_n contains all upper and all lower generalized triangular matrix rings and is called the class of n-by-n trivial generalized matrix rings. This paper is primarily an expository paper based on a plenary talk presented at the 24th International Conference on Nearrings, Nearfields, and Related Topics. However some new results are presented at the end of the paper.