QUASI-FROBENIUS BIMODULES OF FUNCTIONS ON A SEMIGROUP

Abstract

Let _AM_B be a QF-bimodule, A a left Artinian ring, B a right Artinian ring, G a semigroup with a unit element (a monoid). Let M^G be the set of all functions on G with values in M. Consider M^G as an (AG, BG)-bimodule over the semigroup rings AG and BG. It is proved that the annihilator maps I → r_M^G (I) and R → l_AG (R) are mutually inverse bijective Galois correspondences between the set of finitely cogenerated left ideals I ⊆ AG and the set of right BG-submodules R ⊆ M^G finitely generated over B. The maps J → l_M^G (J) and L → r_AG (L) are mutually inverse bijective Galois correspondences between the set of finitely cogenerated right ideals J ⊆ AG and the set of left AG-submodules L ⊆ M^G finitely generated over A. This result also makes it possible, starting from a given QF-bimodule _A M_B , to construct new QF-bimodules _AG/IS_BG/J as bimodules of functions on a semigroup with values in M.

Acknowledgments