nLab artinian ring

Contents

Contents

Definition

Artinian rings

Every ring RR has a canonical RR-RR-bimodule structure, with left action α L:R×RR\alpha_L:R \times R \to R and right action α R:R×RR\alpha_R:R \times R \to R defined as the multiplicative binary operation on RR and biaction α:R×R×RR\alpha:R \times R \times R \to R defined as the ternary product on RR:

α L(a,b)ab\alpha_L(a, b) \coloneqq a \cdot b
α R(a,b)ab\alpha_R(a, b) \coloneqq a \cdot b
α(a,b,c)abc\alpha(a, b, c) \coloneqq a \cdot b \cdot c

Let TwoSidedIdeals(R)\mathrm{TwoSidedIdeals}(R) be the category of two-sided ideals in RR, whose objects are two-sided ideals II in RR, sub- R R - R R -bimodules of RR with respect to the canonical bimodule structure on RR, and whose morphisms are RR-RR-bimodule monomorphisms.

A descending chain of two-sided ideals in RR is an inverse sequence of two-sided ideals in RR, a sequence of two-sided ideals A:TwoSidedIdeals(R)A:\mathbb{N} \to \mathrm{TwoSidedIdeals}(R) with the following dependent sequence of RR-RR-bimodule monomorphisms: for natural number nn \in \mathbb{N}, a dependent RR-RR-bimodule monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n}.

A ring RR is Artinian if it satisfies the descending chain condition on its two-sided ideals: for every descending chain of two-sided ideals (A,i n)(A, i_n) in RR, there exists a natural number mm \in \mathbb{N} such that for all natural numbers nmn \geq m, the RR-RR-bimodule monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n} is an RR-RR-bimodule isomorphism.

Left Artinian rings

Let LeftIdeals(R)\mathrm{LeftIdeals}(R) be the category of left ideals in RR, whose objects are left ideals II in RR, sub-left-RR-modules of RR with respect to the canonical left module structure ()():R×RR(-)\cdot(-):R \times R \to R on RR, and whose morphisms are left RR-module monomorphisms.

A descending chain of left ideals in RR is an inverse sequence of left ideals in RR, a sequence of left ideals A:LeftIdeals(R)A:\mathbb{N} \to \mathrm{LeftIdeals}(R) with the following dependent sequence of left RR-module monomorphisms: for natural number nn \in \mathbb{N}, a dependent left RR-module monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n}.

A ring RR is left Artinian if it satisfies the descending chain condition on its left ideals: for every descending chain of left ideals (A,i n)(A, i_n) in RR, there exists a natural number mm \in \mathbb{N} such that for all natural numbers nmn \geq m, the left RR-module monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n} is an left RR-module isomorphism.

Right Artinian rings

Let RightIdeals(R)\mathrm{RightIdeals}(R) be the category of right ideals in RR, whose objects are right ideals II in RR, sub-right-RR-modules of RR with respect to the canonical right module structure ()():R×RR(-)\cdot(-):R \times R \to R on RR, and whose morphisms are right RR-module monomorphisms.

A descending chain of right ideals in RR is an inverse sequence of right ideals in RR, a sequence of right ideals A:RightIdeals(R)A:\mathbb{N} \to \mathrm{RightIdeals}(R) with the following dependent sequence of right RR-module monomorphisms: for natural number nn \in \mathbb{N}, a dependent right RR-module monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n}.

A ring RR is right Artinian if it satisfies the descending chain condition on its right ideals: for every descending chain of right ideals (A,i n)(A, i_n) in RR, there exists a natural number mm \in \mathbb{N} such that for all natural numbers nmn \geq m, the right RR-module monomorphism i n:A n+1A ni_n:A_{n+1} \hookrightarrow A_{n} is an right RR-module isomorphism.

Properties

In an artinian ring RR the Jacobson radical J(R)J(R) is nilpotent. A left artinian ring is semiprimitive if and only if the zero ideal is the unique nilpotent ideal.

Artinian and Noetherian rings

A dual condition is noetherian: a noetherian ring is a ring satisfying the ascending chain condition on ideals. The symmetry is severely broken if one considers unital rings: for example every unital artinian ring is noetherian. For a converse there is a strong condition: a left (unital) ring RR is left artinian iff R/J(R)R/J(R) is semisimple in RMod_R Mod and the Jacobson radical J(R)J(R) is nilpotent. Artinian rings are intuitively much smaller than generic noetherian rings.

See also

Last revised on May 25, 2022 at 22:14:39. See the history of this page for a list of all contributions to it.