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.

In connection with the fact that in the infinite-dimensional case the largest nilpotent ideal need not exist, many different generalizations of the classical ideal have appeared: the Baer radical, the Jacobson radical, the Levitzki radical, the Köthe radical, etc. The most frequently used of these is the Jacobson radical (cf. also Quasi-regular radical). Radicals that are in some sense opposite to the classical radical have also been introduced. For example, all classical semi-simple rings (that is, direct sums of complete matrix rings) are radical in the sense of the regular von Neumann radical and are hereditary for the idempotent Baer radical. The construction of the general theory of radicals was initiated by S. Amitsur

and A.G. Kurosh [2].

General theory of radicals.

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.

Hereditary radicals.

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 [5], [7]).

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.

References

[1a]	S.A. Amitsur, "A general theory of radicals I. Radicals in complete lattices" Amer. J. Math. , 74 (1952) pp. 774–786
[1b]	S.A. Amitsur, "A general theory of radicals II. Radicals in rings and bicategories" Amer. J. Math. , 76 (1954) pp. 100–125
[1c]	S.A. Amitsur, "A general theory of radicals III. Applications" Amer. J. Math. , 76 (1954) pp. 126–136
[2]	A.G. Kurosh, "Radicals of rings and algebras" Mat. Sb. , 33 : 1 (1953) pp. 13–26 (In Russian)
[3]	N.J. Divinsky, "Rings and radicals" , Allen & Unwin (1965)
[4]	E. Artin, C.J. Nesbitt, R.M. Thrall, "Rings with minimum condition" , Univ. Michigan Press , Ann Arbor (1946)
[5]	Itogi Nauk. Algebra Topol. Geom. 1967 (1969) pp. 28–32
[6]	V.A. Andrunakievich, Yu.M. Rabukhin, "The theory of radicals of rings" L.A. Bokut' (ed.) et al. (ed.) , Rings , 2 , Novosibirsk (1973) pp. 3–6 (In Russian)
[7]	V.A. Andrunakievich, Yu.M. Ryabukhin, "Radicals of algebras and structure theory" , Moscow (1979) (In Russian)
[8]	K.A. Zhevlakov, A.M. Slin'ko, I.P. Shestakov, A.I. Shirshov, "Rings that are nearly associative" , Acad. Press (1982) (Translated from Russian)

V.A. Andrunakievich

In the class of Lie algebras the radical is the largest solvable ideal, that is, the solvable ideal $ \mathfrak r $ containing all solvable ideals of the given Lie algebra (cf. Solvable group). In a finite-dimensional Lie algebra $ \mathfrak g $ there is also a largest nilpotent ideal $ \mathfrak n $( sometimes called the nil radical) that coincides with the largest ideal consisting of nilpotent elements, and also with the set of $ x \in \mathfrak g $ such that the adjoint operator is contained in the radical of the associative algebra of linear transformations of $ \mathfrak g $ generated by the adjoint Lie algebra $ \mathop{\rm ad} \mathfrak g $. The nilpotent radical $ \mathfrak s $ of a Lie algebra $ \mathfrak g $ has also been considered — the set of those $ x \in \mathfrak g $ such that $ \sigma ( x) = 0 $ for any irreducible finite-dimensional linear representation $ \sigma $ of $ \mathfrak g $. The nilpotent radical also coincides with the largest ideal represented by nilpotent operators for any finite-dimensional linear representation of $ \mathfrak g $. Here $ \mathfrak r \supseteq \mathfrak n \supseteq \mathfrak s $. If the characteristic of the ground field is $ 0 $, then $ \mathfrak s $ is the smallest ideal $ \mathfrak i \subset \mathfrak g $ for which $ \mathfrak g / \mathfrak i $ is a reductive Lie algebra (cf. Lie algebra, reductive). In this case the nilpotent radical is related to $ \mathfrak r $ by:

$$ \mathfrak s = [ \mathfrak g , \mathfrak r ] = [ \mathfrak g , \mathfrak g ] \cap \mathfrak r ; $$

any derivation of $ \mathfrak g $ transforms $ \mathfrak r $ to $ \mathfrak n $ and $ \mathfrak s $ to $ \mathfrak s $. The nil radical and the nilpotent radical, however, are not radicals in the sense of the general theory of radicals of rings and algebras.

References

[1]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[2]	, Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Secr. Math. Univ. Paris (1955)
[3]	C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946)

A.L. Onishchik

References

[a1]	N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1960) pp. Chapt. I: Algèbres de Lie