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...

in abstract algebra Radical of a ring, an ideal of "bad" elements of a ring Jacobson radical, consisting of those elements in a ring R that annihilate all...

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...

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...

In ring theory, a branch of mathematics, the radical of an ideal I {\displaystyle I} of a commutative ring is another ideal defined by the property that...

more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as...

integers. Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings (group rings, division...

and radical ideals is a bijection. The coordinate ring of an affine algebraic set is reduced (nilpotent-free), as an ideal I in a ring R is radical if...

Local rings also do not have such idempotents, but for a different reason. The only idempotent contained in the Jacobson radical of a ring is 0. A ring in...

Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements...

Retrieved 2013-10-14. "Pagani Zonda breaks 'Ring record". Evo. 13 November 2018. Retrieved 2024-05-13. "2007 Radical SR8LM - Images, Specifications and Information"...

the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense...

of all nilpotent elements forms an ideal known as the nil radical of a ring. Because the nil radical contains every nilpotent element, an ideal of a commutative...

Jacobson radical is zero. If an Artinian semisimple ring contains a field as a central subring, it is called a semisimple algebra. For a commutative ring, the...

specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring. In this article...

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

algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general...

radicals highly chemically reactive. Many radicals spontaneously dimerize. Most organic radicals have short lifetimes. A notable example of a radical...

ideal and with the ring's Jacobson radical. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal...

unnecessary. A semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general...

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...

theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for...

In organic chemistry, the phenyl group, or phenyl ring, is a cyclic group of atoms with the formula C6H5, and is often represented by the symbol Ph (archaically...

a radical clock is a chemical compound that assists in the indirect methodology to determine the kinetics of a free-radical reaction. The radical-clock...

In organic chemistry, a radical-substitution reaction is a substitution reaction involving free radicals as a reactive intermediate. The reaction always...

involving free-radicals, many-membered rings, and organometallic compounds, as well as the development of Ziegler–Natta catalyst. One of many awards Ziegler...

x n ] {\displaystyle k[x_{1},x_{2},\ldots ,x_{n}]} is a polynomial ring, the ring of radical polynomials is the subring generated by the polynomial ∑...

aryl radical in organic chemistry is a reactive intermediate and an arene compound incorporating one free radical carbon atom as part of the ring structure...

seen as a generalisation of rings. Many concepts from ring theory, such as ideals, Jacobson radicals, and factor rings can be generalized in a straightforward...

precisely of the elements annihilated by the Jacobson radical of R. In the context of Lie algebras, a socle of a symmetric Lie algebra is the eigenspace of its...