Radical of a ring

Last updated February 01, 2024

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 ofR 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).

Related Research Articles

In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left–right symmetric. The Jacobson radical of a ring is frequently denoted by J(R) or rad(R); the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in Jacobson 1945.

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In ring theory, a branch of mathematics, the radical of an ideal $of a commutative ring is another ideal defined by the property that an element is in the radical if and only if some power of is in . Taking the radical of an ideal is called radicalization . A radical ideal is an ideal that is equal to its radical. The radical of a primary ideal is a prime ideal.$

In mathematics, an element $of a ring is called nilpotent if there exists some positive integer, called the index, such that .$

In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements:

In mathematics, specifically abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.

In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x² = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced.

In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yⁿ is also an element of Q, for some n > 0. For example, in the ring of integers Z, (pⁿ) is a primary ideal if p is a prime number.

In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2022. It is formulated in various ways. Suppose that R is a ring. One way to state the conjecture is that if R has no nil ideal, other than {0}, then it has no nil one-sided ideal, other than {0}.

In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent.

In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Elements of a Ore extension are called Ore polynomials.

In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals.

In mathematics, more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that I^k = 0. By I^k, it is meant the additive subgroup generated by the set of all products of k elements in I. Therefore, I is nilpotent if and only if there is a natural number k such that the product of any k elements of I is 0.

In mathematics, more specifically ring theory and the theory of nil ideals, Levitzky's theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent. Levitzky's theorem is one of the many results suggesting the veracity of the Köthe conjecture, and indeed provided a solution to one of Köthe's questions as described in. The result was originally submitted in 1939 as, and a particularly simple proof was given in.

In mathematics, a Goldman domain or G-domain is an integral domain A whose field of fractions is a finitely generated algebra over A. They are named after Oscar Goldman.

In abstract algebra, a uniserial moduleM is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N₁ and N₂ of M, either $or . A module is called a serial module if it is a direct sum of uniserial modules. A ring R is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself. Left uniserial and left serial rings are defined in an analogous way, and are in general distinct from their right counterparts.$

In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in R. In set notation it is usually denoted as $. For general rings, is a good generalization of the torsion submodule tors(M) which is most often defined for domains. In the case that R is a commutative domain, .$

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.