Jacobson radical

In mathematics, more specifically ring theory, a branch of abstract algebra, 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); however to avoid confusion with other radicals of rings, the former notation will be preferred in this article. 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 rings without unity. 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.

Intuitive discussion

As with other radicals of rings, the Jacobson radical can be thought of as a collection of "bad" elements. In this case the "bad" property is that these elements annihilate all simple left and right modules of the ring. For purposes of comparison, consider the nilradical of a commutative ring, which consists of all elements that are nilpotent. In fact for any ring, the nilpotent elements in the center of the ring are also in the Jacobson radical.[1] So, for commutative rings, the nilradical is contained in the Jacobson radical.

The Jacobson radical is very similar to the nilradical in an intuitive sense. A weaker notion of being bad, weaker than being a zero divisor, is being a non-unit (not invertible under multiplication). The Jacobson radical of a ring consists of elements that satisfy a stronger property than being merely a non-unit – in some sense, a member of the Jacobson radical must not "act as a unit" in any module "internal to the ring." More precisely, a member of the Jacobson radical must project under the canonical homomorphism to the zero of every "right division ring" (each non-zero element of which has a right inverse) internal to the ring in question. Concisely, it must belong to every maximal right ideal of the ring. These notions are of course imprecise, but at least explain why the nilradical of a commutative ring is contained in the ring's Jacobson radical.

In yet a simpler way, we may think of the Jacobson radical of a ring as method to "mod out bad elements" of the ring – that is, members of the Jacobson radical act as 0 in the quotient ring, R/J(R). If N is the nilradical of commutative ring R, then the quotient ring R/N has no nilpotent elements. Similarly for any ring R, the quotient ring has J(R/J(R))={0} and so all of the "bad" elements in the Jacobson radical have been removed by modding out J(R). Elements of the Jacobson radical and nilradical can be therefore seen as generalizations of 0.

Equivalent characterizations

The Jacobson radical of a ring has various internal and external characterizations. The following equivalences appear in many noncommutative algebra texts such as (Anderson 1992, §15), (Isaacs 1993, §13B), and (Lam 2001, Ch 2).

The following are equivalent characterizations of the Jacobson radical in rings with unity (characterizations for rings without unity are given immediately afterward):

For rings without unity it is possible for R=J(R), however the equation that J(R/J(R))={0} still holds. The following are equivalent characterizations of J(R) for rings without unity appear in (Lam 2001, p. 63):



See also


  1. 1 2 3 Isaacs, p. 181.
  2. 1 2 3 Isaacs, p. 182.
  3. Isaacs, Problem 12.5, p. 173
  4. Isaacs, Corollary 13.4, p. 180


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