nLab artinian ring

Definition

Artinian rings

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

$$\alpha_L(a, b) \coloneqq a \cdot b$$

$$\alpha_R(a, b) \coloneqq a \cdot b$$

$$\alpha(a, b, c) \coloneqq a \cdot b \cdot c$$

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

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

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

Left Artinian rings

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

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

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

Right Artinian rings

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

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

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

Properties

In an artinian ring $R$ the Jacobson radical $J(R)$ 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 $R$ is left artinian iff $R/J(R)$ is semisimple in $_R Mod$ and the Jacobson radical $J(R)$ is nilpotent. Artinian rings are intuitively much smaller than generic noetherian rings.

See also

• Artinian bimodule

• artinian object?

commutative ring	reduced ring	integral domain
local ring	reduced local ring	local integral domain
Artinian ring	semisimple ring	field
Weil ring	field	field

References

• The Stacks Project

  • 10.100. Flatness criteria over Artinian rings