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 right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left–right symmetric. The Jacobson radical of a ring is frequently denoted by J(R) or rad(R); the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in Jacobson 1945.
The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to non-unital rings. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring- and module-theoretic results, such as Nakayama's lemma.
In mathematics, more specifically ring theory, the Jacobsonradical of a ring R is the ideal consisting of those elements in R that annihilate all simple...
years several other radicals were discovered, of which the most important example is the Jacobsonradical. The general theory of radicals was defined independently...
Krull (1951, 1952), who named them after Nathan Jacobson because of their relation to Jacobsonradicals, and by Oscar Goldman (1951), who named them Hilbert...
properties, a ring is semisimple if and only if it is Artinian and its Jacobsonradical is zero. If an Artinian semisimple ring contains a field as a central...
modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobsonradical for rings....
of all prime ideals of the quotient ring. This is contained in the Jacobsonradical, which is the intersection of all maximal ideals, which are the kernels...
important concept in abstract algebra Radical of a ring, an ideal of "bad" elements of a ring Jacobsonradical, consisting of those elements in a ring...
In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobsonradical is zero. This is a type of ring more...
and unique maximal two-sided ideal of the ring, and is in fact the Jacobsonradical J(R). It is possible for a ring to have a unique maximal two-sided...
algebra over a field which has trivial Jacobsonradical (only the zero element of the algebra is in the Jacobsonradical). If the algebra is finite-dimensional...
unnecessary. A semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobsonradical is zero. This is a type of ring more...
that A is Artinian simplifies the notion of a Jacobsonradical; for an Artinian ring, the Jacobsonradical of A is the intersection of all (two-sided) maximal...
observing that any nil ideal is contained in the Jacobsonradical of the ring, and since the Jacobsonradical is a nilpotent ideal (due to the Artinian hypothesis)...
prime ideal, the Jacobsonradical — which is the intersection of maximal ideals — must contain the nilradical. A ring R is called a Jacobson ring if the nilradical...
observing that any nil ideal is contained in the Jacobsonradical of the ring, and since the Jacobsonradical is a nilpotent ideal (due to the artinian hypothesis)...
coincides with the unique maximal right ideal and with the ring's Jacobsonradical. The third of the properties listed above says that the set of non-units...
necessarily vice versa. Jacobson 1. The Jacobsonradical of a ring is the intersection of all maximal left ideals. 2. A Jacobson ring is a ring in which...
with the nilradical when commutativity is assumed. The concept of the Jacobsonradical of a ring; that is, the intersection of all right (left) annihilators...
intersection of all prime ideals. A characteristic similar to that of Jacobsonradical and annihilation of simple modules is available for nilradical: nilpotent...
such that the product of first n terms are zero), where J(R) is the Jacobsonradical of R. (Bass' Theorem P) R satisfies the descending chain condition...
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