Köthe conjectureIn mathematics, the Köthe conjecture is a problem in ring theory, open . 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}. This question was posed in 1930 by Gottfried Köthe (1905–1989). The Köthe conjecture has been shown to be true for various classes of rings, such as polynomial identity rings and right Noetherian rings, but a general solution remains elusive.
Radical of a ringIn 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 , based on a suggestion of . 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 and . In the theory of radicals, rings are usually assumed to be associative, but need not be commutative and need not have a multiplicative identity.
Nilpotent idealIn 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. The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings.
Idempotent (ring theory)In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that a = a2 = a3 = a4 = ... = an for any positive integer n. For example, an idempotent element of a matrix ring is precisely an idempotent matrix. For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring.
Nilradical of a ringIn algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements: It is thus the radical of the zero ideal. If the nilradical is the zero ideal, the ring is called a reduced ring. The nilradical of a commutative ring is the intersection of all prime ideals. In the non-commutative ring case the same definition does not always work. This has resulted in several radicals generalizing the commutative case in distinct ways; see the article Radical of a ring for more on this.
NilpotentIn mathematics, an element of a ring is called nilpotent if there exists some positive integer , called the index (or sometimes the degree), such that . The term, along with its sister idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras. This definition can be applied in particular to square matrices. The matrix is nilpotent because . See nilpotent matrix for more. In the factor ring , the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9.
Jacobson radicalIn mathematics, more specifically ring theory, the Jacobson radical of a ring is the ideal consisting of those elements in that annihilate all simple right -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 or ; the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring.
Ring theoryIn 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 (group rings, division rings, universal enveloping algebras), 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.
