Artin module

An Artin module is a module above a ring in which the following condition for breaking a decreasing chain is fulfilled . Symbolically, the module $M$ ${\ displaystyle M}$ $M$ Artins, if any sequence of its submodules:

M_{1}\supset M_{2}\supset \ldots \supset M_{i}\supset \ldots

{\ displaystyle M_ {1} \ supset M_ {2} \ supset \ ldots \ supset M_ {i} \ supset \ ldots}

{\ displaystyle M_ {1} \ supset M_ {2} \ supset \ ldots \ supset M_ {i} \ supset \ ldots}

stabilizes, i.e. starting from some $n$ ${\ displaystyle n}$ $n$ done:

M_{n}=M_{n+1}=\ldots

{\ displaystyle M_ {n} = M_ {n + 1} = \ ldots}

{\ displaystyle M_ {n} = M_ {n + 1} = \ ldots}

.

This statement is equivalent to the fact that in any nonempty set of submodules $M$ ${\ displaystyle M}$ $M$ there is a minimal element .

If a $M$ ${\ displaystyle M}$ $M$ - Artins, then any of its submodules and any of its factor modules are Artinian. Conversely, if the submodule $N\subset M$ ${\ displaystyle N \ subset M}$ $N \ subset M$ and factor module $M/N$ ${\ displaystyle M / N}$ $M / N$ Artinian, then the module itself $M$ ${\ displaystyle M}$ $M$ 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 $A$ ${\ displaystyle A}$ $A$ with a single element is called Artinian if it is Artinian $A$ ${\ displaystyle A}$ $A$ -module (satisfies the condition of breaking off decreasing chains for ideals , for the non-commutative case, respectively, left or right ).

Literature

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.

Artin module

Literature

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