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In mathematics, a Lie group is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be a group, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group.
In ring theory, a branch of abstract algebra, 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 noncommutative rings where multiplication is not required to be commutative.
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants that differ in the additional conditions imposed on the orders of the group elements, some of which are still open questions.
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 mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index, such that xn = 0.
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.
In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good" elements of the ring.
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. Equivalently, a domain is a ring in which 0 is the only left zero divisor. A commutative domain is called an integral domain. Mathematical literature contains multiple variants of the definition of "domain".
In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated.
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}.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which each element has finite order. The exponent of such a group, if extant, is the least common multiple of the orders of the elements. The exponent exists for any finite group, and it divides the group order.
In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group.
A non-associative algebra is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), d and a(b ) may all yield different answers.
In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite.
In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem.
Evgenii Solomonovich Golod was a Russian mathematician who proved the Golod–Shafarevich theorem on class field towers. As an application, he gave a negative solution to the Kurosh–Levitzky problem on the nilpotency of finitely generated nil algebras, and so to a weak form of Burnside's problem.
Sergei Nikolaevich Chernikov was a Russian mathematician who contributed significantly to the development of infinite group theory and linear inequalities.
References
- Vesselin S. Drensky, Edward Formanek (2004), Polynomial Identity Rings, p. 89.
- Some open problems in the theory of infinite dimensional algebras (2007). E. Zelmanov.
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