radical theory

Let $𝒳$ represent a property which a ring may or may not have. This property may be anything at all: what is important is that for any ring $R$, the statement “$R$ has property $𝒳$” is either true or false.

We say that a ring which has the property $𝒳$ is an $𝒳$-ring. An ideal $I$ of a ring $R$ is called an $𝒳$-ideal if, as a ring, it is an $𝒳$-ring. (Note that this definition only makes sense if rings are not required to have identity elements; otherwise and ideal is not, in general, a ring. Rings are not required to have an identity element in radical theory.)

The property $𝒳$ is a radical property if it satisfies:

1. The class of $𝒳$-rings is closed under homomorphic images.

2. Every ring $R$ has a largest $𝒳$-ideal, which contains all other $𝒳$-ideals of $R$. This ideal is written $𝒳(R)$.

3. $𝒳(R/𝒳(R))=0$.

The ideal $𝒳(R)$ is called the $\mathrm{X}$-radical of $R$. A ring is called $\mathrm{X}$-radical if $𝒳(R)=R$, and is called $\mathrm{X}$-semisimple if $𝒳(R)=0$.

If $𝒳$ is a radical property, then the class of $𝒳$-rings is also called the class of $\mathrm{X}$-radical rings.

The class of $𝒳$-radical rings is closed under ideal extensions. That is, if $A$ is an ideal of $R$, and $A$ and $R/A$ are $𝒳$-radical, then so is $R$.

Radical theory is the study of radical properties and their interrelations. There are several well-known radicals which are of independent interest in ring theory (See examples – to follow).

The class of all radicals is however very large. Indeed, it is possible to show that any partition of the class of simple rings into two classes $ℛ$ and $𝒮$ such that isomorphic simple rings are in the same class, gives rise to a radical $𝒳$ with the property that all rings in $ℛ$ are $𝒳$-radical and all rings in $𝒮$ are $𝒳$-semisimple. In fact, there are at least two distinct radicals for each such partition.

A radical $𝒳$ is hereditary if every ideal of an $𝒳$-radical ring is also $𝒳$-radical.

A radical $𝒳$ is supernilpotent if the class of $𝒳$-rings contains all nilpotent rings.

1 Examples

Nil is a radical property. This property defines the nil radical, $𝒩$.

Nilpotency is not a radical property.

Quasi-regularity is a radical property. The associated radical is the Jacobson radical, $𝒥$.

Title	radical theory
Canonical name	RadicalTheory
Date of creation	2013-03-22 13:13:02
Last modified on	2013-03-22 13:13:02
Owner	mclase (549)
Last modified by	mclase (549)
Numerical id	10
Author	mclase (549)
Entry type	Definition
Classification	msc 16N80
Related topic	JacobsonRadical
Defines	radical
Defines	radical property
Defines	semisimple
Defines	hereditary
Defines	hereditary radical
Defines	supernilpotent
Defines	supernilpotent radical