In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.
Edward Witten is an American mathematical and theoretical physicist. He is a professor emeritus in the school of natural sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. Witten's work has also significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, for his mathematical insights in physics, such as his 1981 proof of the positive energy theorem in general relativity, and his interpretation of the Jones invariants of knots as Feynman integrals. He is considered the practical founder of M-theory.
Amalie Emmy Noether was a German mathematician who made many important contributions to abstract algebra. She discovered Noether's First and Second Theorem, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed some theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.
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.
Daniel Gray "Dan" Quillen was an American mathematician. He is known for being the "prime architect" of higher algebraic K-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978.
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.
Irving Kaplansky was a mathematician, college professor, author, and amateur musician.
The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures.
In mathematics, a Drinfeld module is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka is a sort of generalization of a Drinfeld module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it.
In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively non-negative) on that set. Precisely, Let p be a polynomial in n variables with real coefficients and let S be a subset of the n-dimensional Euclidean space ℝn. We say that:
In algebra, the term Hermite ring has been applied to three different objects.
Claudio Procesi is an Italian mathematician, known for works in algebra and representation theory.
Tamar Debora Ziegler is an Israeli mathematician known for her work in ergodic theory, combinatorics and number theory. She holds the Henry and Manya Noskwith Chair of Mathematics at the Einstein Institute of Mathematics at the Hebrew University.
In mathematics, a clean ring is a ring in which every element can be written as the sum of a unit and an idempotent. A ring is a local ring if and only if it is clean and has no idempotents other than 0 and 1. The endomorphism ring of a continuous module is a clean ring. Every clean ring is an exchange ring. A matrix ring over a clean ring is itself clean.
Henri Moscovici is a Romanian-American mathematician, specializing in non-commutative geometry and global analysis.
Iain Gordon, FRSE, is a mathematician, currently Iain Gordon is Professor of Mathematics, Vice-Principal and Head of the College of Science and Engineering at the University of Edinburgh. His field of specialisation is representation theory and noncommutative algebra.
Wallace Smith Martindale III is an American mathematician, known for Martindale's Theorem (1969) and the Martindale ring of quotients introduced in the proof of the theorem. His 1969 paper generalizes Posner's theorem and a theorem of Amitsur and gives an independent, unified proof of the two theorems.
Klaus Wilhelm Roggenkamp was a German mathematician, specializing in algebra.
