Radical of a ring

In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good" elements of the ring.

The first example of a radical was the nilradical introduced by Köthe (1930), based on a suggestion of Wedderburn (1908). In the next few years several other radicals were discovered, of which the most important example is the Jacobson radical. The general theory of radicals was defined independently by (Amitsur 1952, 1954, 1954b) and Kurosh (1953).

Definitions

In the theory of radicals, rings are usually assumed to be associative, but need not be commutative and need not have a multiplicative identity. In particular, every ideal in a ring is also a ring.

A radical class (also called radical property or just radical) is a class σ of rings possibly without multiplicative identities, such that:

the homomorphic image of a ring in σ is also in σ
every ring R contains an ideal S(R) in σ that contains every other ideal of R that is in σ
S(R/S(R)) = 0. The ideal S(R) is called the radical, or σ-radical, of R.

The study of such radicals is called torsion theory.

For any class δ of rings, there is a smallest radical class Lδ containing it, called the lower radical of δ. The operator L is called the lower radical operator.

A class of rings is called regular if every non-zero ideal of a ring in the class has a non-zero image in the class. For every regular class δ of rings, there is a largest radical class Uδ, called the upper radical of δ, having zero intersection with δ. The operator U is called the upper radical operator.

A class of rings is called hereditary if every ideal of a ring in the class also belongs to the class.

Examples

The Jacobson radical

Let R be any ring, not necessarily commutative. The Jacobson radical of R is the intersection of the annihilators of all simple right R-modules.

There are several equivalent characterizations of the Jacobson radical, such as:

J(R) is the intersection of the regular maximal right (or left) ideals of R.
J(R) is the intersection of all the right (or left) primitive ideals of R.
J(R) is the maximal right (or left) quasi-regular right (resp. left) ideal of R.

As with the nilradical, we can extend this definition to arbitrary two-sided ideals I by defining J(I) to be the preimage of J(R/I) under the projection map R → R/I.

If R is commutative, the Jacobson radical always contains the nilradical. If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I. This says that R is a Jacobson ring.

The Baer radical

The Baer radical of a ring is the intersection of the prime ideals of the ring R. Equivalently it is the smallest semiprime ideal in R. The Baer radical is the lower radical of the class of nilpotent rings. Also called the "lower nilradical" (and denoted Nil_∗R), the "prime radical", and the "Baer-McCoy radical". Every element of the Baer radical is nilpotent, so it is a nil ideal.

For commutative rings, this is just the nilradical and closely follows the definition of the radical of an ideal.

The upper nil radical or Köthe radical

The sum of the nil ideals of a ring R is the upper nilradical Nil^*R or Köthe radical and is the unique largest nil ideal of R. Köthe's conjecture asks whether any left nil ideal is in the nilradical.

Singular radical

An element of a (possibly non-commutative ring) is called left singular if it annihilates an essential left ideal, that is, r is left singular if Ir = 0 for some essential left ideal I. The set of left singular elements of a ring R is a two-sided ideal, called the left singular ideal, and is denoted ${\mathcal {Z}}(_{R}R)$ . The ideal N of R such that $N/{\mathcal {Z}}(_{R}R)={\mathcal {Z}}(_{R/{\mathcal {Z}}(_{R}R)}R/{\mathcal {Z}}(_{R}R))\,$ is denoted by ${\mathcal {Z}}_{2}(_{R}R)$ and is called the singular radical or the Goldie torsion of R. The singular radical contains the prime radical (the nilradical in the case of commutative rings) but may properly contain it, even in the commutative case. However, the singular radical of a Noetherian ring is always nilpotent.

The Levitzki radical

The Levitzki radical is defined as the largest locally nilpotent ideal, analogous to the Hirsch–Plotkin radical in the theory of groups. If the ring is Noetherian, then the Levitzki radical is itself a nilpotent ideal, and so is the unique largest left, right, or two-sided nilpotent ideal.^{[citation needed]}

The Brown–McCoy radical

The Brown–McCoy radical (called the strong radical in the theory of Banach algebras) can be defined in any of the following ways:

the intersection of the maximal two-sided ideals
the intersection of all maximal modular ideals
the upper radical of the class of all simple rings with multiplicative identity

The Brown–McCoy radical is studied in much greater generality than associative rings with 1.

The von Neumann regular radical

A von Neumann regular ring is a ring A (possibly non-commutative without multiplicative identity) such that for every a there is some b with a = aba. The von Neumann regular rings form a radical class. It contains every matrix ring over a division algebra, but contains no nil rings.

The Artinian radical

The Artinian radical is usually defined for two-sided Noetherian rings as the sum of all right ideals that are Artinian modules. The definition is left-right symmetric, and indeed produces a two-sided ideal of the ring. This radical is important in the study of Noetherian rings, as outlined by Chatters & Hajarnavis (1980).

References

Amitsur, S. A. (1952). "A general theory of radicals. I: Radicals in complete lattices". American Journal of Mathematics. 74: 774–786. doi:10.2307/2372225. JSTOR 2372225.
Amitsur, S. A. (1954). "A general theory of radicals. II: Radicals in rings and bicategories". American Journal of Mathematics. 75: 100–125. doi:10.2307/2372403. JSTOR 2372403.
Amitsur, S. A. (1954b). "A general theory of radicals. III: Applications". American Journal of Mathematics. 75: 126–136. doi:10.2307/2372404. JSTOR 2372404.
Chatters, A. W.; Hajarnavis, C. R. (1980), Rings with Chain Conditions, Research Notes in Mathematics, vol. 44, Boston, Massachusetts: Pitman (Advanced Publishing Program), pp. vii+197, ISBN 0-273-08446-1, MR 0590045
Köthe, Gottfried (1930). "Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständig reduzibel ist". Mathematische Zeitschrift. 32 (1): 161–186. doi:10.1007/BF01194626. S2CID 123292297.
Kurosh, A. G. (1953). "Radicals of rings and algebras". Matematicheskii Sbornik (in Russian). 33: 13–26.
Wedderburn, J.H.M. (1908). "On Hypercomplex Numbers". Proceedings of the London Mathematical Society. 6: 77–118. doi:10.1112/plms/s2-6.1.77.