An Artin module is a module above a ring in which the following condition for breaking a decreasing chain is fulfilled . Symbolically, the module Artins, if any sequence of its submodules:
stabilizes, i.e. starting from some done:
This statement is equivalent to the fact that in any nonempty set of submodules there is a minimal element .
If a - Artins, then any of its submodules and any of its factor modules are Artinian. Conversely, if the submodule and factor module Artinian, then the module itself Artins.
They are named after Emil Artin , along with similar general algebraic structures with conditions for breaking off decreasing chains ( Artinian group , Artinian ring ), and dual “Noetherian” structures with a condition for breaking off increasing chains ( Noetherian module , Noetherian group , Noetherian ring ). In particular, the associative ring with a single element is called Artinian if it is Artinian -module (satisfies the condition of breaking off decreasing chains for ideals , for the non-commutative case, respectively, left or right ).
- Atya M., MacDonald I. Introduction to commutative algebra. - M .: Mir, 1972.
- Zarissky O., Samuel R. Commutative algebra. - M: IL, 1963.
- Leng S. Algebra. - M: Mir, 1968.