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Hopkins–Levitzki Theorem
In the branch of abstract algebra called ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in Module (mathematics), modules over semiprimary rings. A ring ''R'' (with 1) is called semiprimary if ''R''/''J''(''R'') is semisimple algebra, semisimple and ''J''(''R'') is a nilpotent ideal, where ''J''(''R'') denotes the Jacobson radical. The theorem states that if ''R'' is a semiprimary ring and ''M'' is an ''R'' module, the three module conditions noetherian module, Noetherian, artinian module, Artinian and "has a composition series" are equivalent. Without the semiprimary condition, the only true implication is that if ''M'' has a composition series, then ''M'' is both Noetherian and Artinian. The theorem takes its current form from a paper by Charles Hopkins and a paper by Jacob Levitzki, both in 1939. For this reason it is often cited as the Hopkins–Levitzki theorem. However Yasuo Akizuki is sometim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
