---
tag: 9ZBM46J88T
kind: statement
role: proposition
name: Artinian reduced rings are semisimple
msc: [16P20, 16K99, 16N20]
uses: [20QEAM8AX2, SZHY975FFX, 0ZB2JG1M9C, NY25HXA478, XA0DCNH7WY, H7RAXJV9CP, R1F271180N, 0EA5BTA0CX, 04C5ZBG1WT]
about: [20QEAM8AX2, SZHY975FFX, NY25HXA478]
sources:
  - url: https://en.wikipedia.org/wiki/Artinian_ring
---

## Statement

Let $R$ be a left [[Artinian ring|20QEAM8AX2]] with identity.
Assume that $R$ is [[reduced|SZHY975FFX]]: its only [[nilpotent element|0ZB2JG1M9C]] is $0$.
Then $R$ is a [[semisimple ring|NY25HXA478]].
More precisely, $R \cong D_1 \times \cdots \times D_k$ for some [[division rings|0EA5BTA0CX]] $D_1, \dots, D_k$.
If $R$ is moreover commutative, then $R$ is a finite product of [[fields|04C5ZBG1WT]].

## Remarks

The [[Jacobson radical|XA0DCNH7WY]] $J$ of a left Artinian ring is [[nilpotent|H7RAXJV9CP]], so each element of $J$ is nilpotent.
Since $R$ is reduced, $J = 0$.
A left Artinian ring with zero Jacobson radical is semisimple, and the [[Artin–Wedderburn theorem|R1F271180N]] gives $R \cong M_{n_1}(D_1) \times \cdots \times M_{n_k}(D_k)$.
If some $n_i \ge 2$, the matrix unit $e_{12}$ in that factor is a nonzero element with square $0$, which is impossible; so every $n_i = 1$.

Example: $\Z/6\Z \cong \Z/2\Z \times \Z/3\Z$ is Artinian and reduced, and it is a product of two fields.
The Artinian hypothesis is needed: $\Z$ is reduced but not semisimple, since the ideal $2\Z$ has no complement in $\Z$.
Reducedness is needed: $\Z/4\Z$ is Artinian, but $2$ is a nonzero nilpotent element, and $\Z/4\Z$ is not semisimple.

## Explanations

### L1 human

An Artinian ring is a ring with a strong finiteness condition on its ideals.
If such a ring has no nonzero element whose power is zero, then it breaks up into finitely many pieces, and each piece is a division ring.
In the commutative case, each piece is a field.

### L2 human

The nilpotent elements are the only obstruction to semisimplicity for Artinian rings, because the radical of an Artinian ring is a nilpotent ideal.
So for Artinian rings, "reduced" implies "semiprimitive", which implies "semisimple".
In commutative algebra this is the statement that a reduced Artinian ring is a finite product of fields.
It is used, for example, to describe reduced zero-dimensional Noetherian rings and finite étale algebras over a field.