17763

Essentially Retractable Modules.

Vedadi M. R. نويسنده

355

360

6

We call a module MR essentially retractable if HomR (M,N ) (not equal) 0 for all essential submodules N of M. For a right FBN ring R, it is shown that: (i) A nonzero module R M is retractable (in the sense that HomR (M,N) (not equal) 0 for all nonzero N =< MR ) if and only if certain factor modules of M are essentially retractable nonsingular modules over R modulo their annihilators. (ii) A non-zero module R M is essentially retractable if and only if there exists a prime ideal P (element of) Ass (MR) such that HomR (M,N) (not equal) 0. Over semiprime right nonsingular rings, a nonsingular essentially retractable module is precisely a module with nonzero dual. Moreover, over certain rings R, including right FBN rings, it is shown that a nonsingular module M with enough uniforms is essentially retractable if and only if there exist uniform retractable R-modules {Ui} i(element of) I and R-homomorphisms M (right arrow) (alpha) i(element of)I Ui(right arrow) (beta) M with (beta)(alpha) (not equal)0.

1201694