---
tag: 2T1AGV1K13
kind: definition
name: Primitive ring
msc: [16D60]
defines: 8F6EHDVJ62
uses: [0894J980RK, 09VDS2W1E8, G00C22AEDH, TB40Y3SNSX]
sources:
  - url: https://en.wikipedia.org/wiki/Primitive_ring
---

## Definition

Let $R$ be a [[ring|0894J980RK]] with identity.
$R$ is *left primitive* if there is a [[simple|G00C22AEDH]] left $R$-[[module|09VDS2W1E8]] $M$ that is [[faithful|TB40Y3SNSX]], that is, $rM = 0$ implies $r = 0$ for $r \in R$.
*Right primitive* rings are defined in the same way with right modules.

## Remarks

Example: let $V$ be a vector space over a field $k$ and $R = \End_k(V)$.
$V$ is a simple $R$-module, since any nonzero vector can be sent to any vector by a linear map.
It is faithful, since a linear map that kills every vector is $0$.
Non-example: $\Z$ is not primitive, since its simple modules are the groups $\Z/p\Z$, and $p$ kills $\Z/p\Z$; no simple module is faithful.
A commutative ring is primitive if and only if it is a field.
Every simple ring with identity is left and right primitive.
A left Artinian ring is left primitive if and only if it is isomorphic to $M_n(D)$ for a division ring $D$ and an integer $n \ge 1$.
There are left primitive rings that are not right primitive.