# Radical of rings and algebras

A concept that first arose in the classical structure theory of finite-dimensional algebras at the beginning of the 20th century. Initially the radical was taken to be the largest nilpotent ideal of a finite-dimensional associative algebra. Algebras with zero radical (called semi-simple) have obtained a fairly complete description in the classical theory: Any semi-simple finite-dimensional associative algebra is a direct sum of simple matrix algebras over suitable fields. Afterwards it was shown that largest nilpotent ideals exist in associative rings and algebras with a minimum condition for left (or right) ideals, that is, in Artinian rings and algebras (cf. Artinian ring), and that the description of Artinian semi-simple rings and algebras coincides with the description of finite-dimensional semi-simple algebras. At the same time it turned out that the radical, as well as the largest nilpotent or largest solvable ideal, could be defined in many classes of finite-dimensional non-associative algebras (alternative, Jordan, Lie, etc.). Here, as in the associative case, semi-simple algebras turned out to be direct sums of simple algebras of some special form.

and A.G. Kurosh .

In this section only algebras (over an arbitrary fixed commutative associative ring with an identity) will be considered; any ring is a special case of such an algebra. By an ideal of an algebra, if not otherwise stipulated, is meant a two-sided ideal.

Let $\mathfrak A$ be a class of algebras that is closed under taking ideals and homomorphic images, that is, containing with each algebra all of its ideals and all of its homomorphic images. Let $r$ be an abstract property that an algebra of $\mathfrak A$ may or may not have. An algebra having property $r$ is called an $r$- algebra. An ideal $I$ of an algebra $A$ is called an $r$- ideal if $I$ is an $r$- algebra. An algebra is called $r$- semi-simple if it has no non-zero $r$- ideals. It is said that $r$ is a radical property of the class $\mathfrak A$, or that there is given a radical in $\mathfrak A$( in the sense of Kurosh), if the following conditions are satisfied:

a) a homomorphic image of an $r$- algebra is an $r$- algebra;

b) each algebra $A$ of $\mathfrak A$ has a largest $r$- ideal, that is, an ideal containing any $r$- ideal of this algebra; this maximal $r$- ideal is then called the $r$- radical of $A$ and is denoted by $r ( A)$;

c) the quotient algebra $A / r ( A)$ is $r$- semi-simple.

An algebra coinciding with its radical is called a radical algebra. In any class of algebras and for any radical, $\{ 0 \}$ is the unique algebra that is simultaneously radical and semi-simple. The subdirect product of any set of semi-simple algebras is itself semi-simple.

Associated with each radical $r$ there are two subclasses of algebras in $\mathfrak A$; the class ${\mathcal R} ( r)$ of all $r$- radical algebras and the class ${\mathcal P} ( r)$ of all $r$- semi-simple algebras. With respect to each of these classes a radical $r ( A)$, for each algebra $A$ from $\mathfrak A$, can be defined, namely:

$$r ( A) = \sum \{ {I } : {I \textrm{ is an ideal in } A ,\ I \in {\mathcal R} ( r) } \} ,$$

respectively,

$$r ( A) = \cap \{ {I } : {I \textrm{ is an ideal in } A ,\ A / I \in {\mathcal P} ( r) } \} .$$

An algebra is $r$- radical if and only if it cannot be mapped homomorphically onto a non-zero $r$- semi-simple algebra.

Necessary and sufficient conditions are known for a subclass of algebras to be the class of all radical or all semi-simple algebras for some radical on $\mathfrak A$. Such subclasses are usually called, respectively, radical or semi-simple subclasses.

The partial ordering of radical classes by inclusion induces a partial order on the class of all radicals on $\mathfrak A$. Namely, $r _ {1} < r _ {2}$ if ${\mathcal R} ( r _ {1} )$ contains ${\mathcal R} ( r _ {2} )$( and, in this case, ${\mathcal P} ( r _ {1} )$ contains ${\mathcal P} ( r _ {2} )$).

For each subclass $M$ of $\mathfrak A$ the lower radical class $l ( M)$ generated by $M$ is the least radical class containing $M$, and the radical corresponding to it is called the lower radical determined by $M$. The upper radical class $u ( M)$ determined by $M$ is the largest radical class relative to the radicals of which all the algebras from $M$ are semi-simple (this radical is called the upper radical determined by $M$). For any class $M$ the lower radical class $l ( M)$ exists. If $\mathfrak A$ is a class of associative algebras, then any subclass $M$ also has an upper radical. In the non-associative case the upper radical need not exist. Sufficient conditions on a class $M$ are known for the upper radical for $M$ to exist. These conditions, in particular, are satisfied by every class containing only simple algebras.

For any radical type a simple algebra is either radical or semi-simple. Thus, corresponding to each radical type there is a partition of the class of simple algebras into two disjoint classes: the class $S _ {1}$ of $r$- semi-simple algebras, or the upper class, and the class $S _ {2}$ of all $r$- radical simple algebras, or the lower class. One says that the radical $r$ corresponds to this partition. Conversely, for an arbitrary partition of the simple algebras into two disjoint classes, one of which, $S _ {1}$, is called upper and the other, $S _ {2}$, is called lower, there is a radical corresponding to the given partition. These will be the upper radical $r _ {1}$ determined by $S _ {1}$, as well as the lower radical $r _ {2}$ determined by $S _ {2}$; the radicals $r _ {1}$ and $r _ {2}$ are called, respectively, the upper and lower radicals of the given partition of the class of simple algebras. For any radical $r$ corresponding to the same partition of the simple algebras, $r _ {1} \geq r \geq r _ {2}$. In the class of all associative algebras, for any partition of the simple algebras, $r _ {1} > r _ {2}$. The classical radical in the class of finite-dimensional associative algebras over a field corresponds to the partition of the simple algebras with empty lower class; moreover, there is a unique non-trivial radical corresponding to this partition.

A radical $r$ is called an ideally hereditary radical, or a torsion radical, in the class $\mathfrak A$ if for any ideal $I$ of an algebra $A$ of this class one has $r ( I) = r ( A) \cap I$. Ideally hereditary radicals are precisely the radicals for which the classes ${\mathcal R} ( r)$ and ${\mathcal P} ( r)$ are closed under passing to ideals. A radical $r$ is called hereditary if the class ${\mathcal R} ( r)$ is closed under passing to ideals. In the class of associative, and also in that of alternative, algebras, each hereditary radical is torsion. A radical $r$ is called strictly hereditary if the class ${\mathcal P} ( r)$ is closed under taking subalgebras.

The class of all torsion radicals is a complete distributive "lattice" (see Distributive lattice). The use of quotation marks here is due to the fact that the collection of elements of this "lattice" is not a set but a class.

In the class of all torsion radicals two opposite subclasses may be distinguished: the class of super-nilpotent torsion radicals, that is, torsion radicals $r$ such that all algebras with zero multiplication are $r$- radical, and the class of sub-idempotent torsion radicals, i.e., torsion radicals $r$ such that all algebras with zero multiplication are $r$- semi-simple (and all $r$- radical algebras are idempotent). An important special case of super-nilpotent radicals are the special radicals, i.e., torsion radicals $r$ such that all $r$- semi-simple algebras decompose into a subdirect sum of primary $r$- semi-simple algebras. There are super-nilpotent non-special radicals (see , ).

## Radicals in the class of associative rings.

Let $\mathfrak A$ be the class of all associative rings and define:

$\phi$— the lower radical determined by the class of all simple rings with zero multiplication;

$\beta$( the lower Baer radical) — the lower radical determined by the class of all nilpotent rings; the upper radical determined by the class of all primary rings; the least special radical; or the intersection of the prime ideals of the ring;

${\mathcal L}$( the Levitzki radical) — the lower radical determined by the class of all locally nilpotent rings; or the sum of all locally nilpotent ideals of the ring and containing every one-sided locally nilpotent ideal of the ring;

${\mathcal K}$( the upper nil radical or Köthe radical) — the lower radical determined by the class of all nil rings;

${\mathcal J}$( the Jacobson radical) — the upper radical determined by the class of all primitive rings; the intersection of all primitive ideals of the ring; or the intersection of all modular maximal right (left) ideals. It is a quasi-regular ideal containing all quasi-regular right (left) ideals;

${\mathcal T}$( the Brown–McCoy radical) — the upper radical determined by the class of all simple rings with an identity. It coincides with the upper radical of its partition; it is equal to the intersection of all maximal modular ideals of the rings;

$\tau$— the upper radical determined by the class of all matrix rings over fields;

${\mathcal A}$( the generalized nil radical) — the upper radical determined by the class of all rings without divisor of zero;

$F$— the upper radical determined by the class of all fields.

In the class of associative rings one has the strict inequalities:

$$\phi < \beta < {\mathcal L} < {\mathcal K} < {\mathcal J} < {\mathcal T} < \tau < F ,$$

$${\mathcal K} < {\mathcal A} < F .$$

In the class of rings with a minimum condition the first seven radicals coincide and correspond to the classical radical. If a radical $r$ induces the radical on the class of rings with a minimum condition, then $\phi < r < \tau$. For rings with a maximum condition, $\beta = {\mathcal L} = {\mathcal K}$. For commutative rings, ${\mathcal J} = {\mathcal T} = \tau = F$, $\beta = {\mathcal L} = {\mathcal K} = {\mathcal A}$. The radicals $\beta$, ${\mathcal L}$, ${\mathcal K}$, ${\mathcal J}$, ${\mathcal T}$, $\tau$, ${\mathcal A}$, $F$ are special. The radicals $\phi$, $\beta$, ${\mathcal L}$ correspond to the same partition of the simple rings, and ${\mathcal J}$, ${\mathcal T}$, $\tau$, ${\mathcal A}$, $F$ to other pairwise different partitions.

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