ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good"[definition needed] elements of the ring. The first example of a radical...

mathematics, more specifically ring theory, the Jacobson radical of a ring R {\displaystyle R} is the ideal consisting of those elements in R {\displaystyle...

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...

Radical of an ideal, an important concept in abstract algebra Radical of a ring, an ideal of "bad" elements of a ring Jacobson radical, consisting of...

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 in one indeterminate) is called the coordinate ring of X. The ideal I ( X ) {\displaystyle \mathrm {I} (X)} is radical, so the coordinate ring is...

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

In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called addition and multiplication, which obey the same...

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

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

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...

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

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

criterion on normality) radical 1.  The Jacobson radical of a ring. 2.  The nilradical of a ring. 3.  A radical of an element x of a ring is an element such...

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...

rings are the most common products; formation of smaller and larger rings is rarely observed. Three conditions must be met for an efficient radical cyclization...

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

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...

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...

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...

radical of a ring. Because the nil radical contains every nilpotent element, an ideal of a commutative ring is nil if and only if it is a subset of the nil...

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...

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...

radical of an ideal I in a polynomial ring with real coefficients is the largest ideal containing I with the same (real) vanishing locus. It plays a similar...

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...

that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the Artinian hypothesis)...

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...