Jacobson Radical - Equivalent Characterizations

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):

• J(R) equals the intersection of all maximal right ideals of the ring. It is also true that J(R) equals the intersection of all maximal left ideals within the ring. These characterizations are internal to the ring, since one only needs to find the maximal right ideals of the ring. For example, if a ring is local, and has a unique maximal right ideal, then this unique maximal right ideal is an ideal because it is exactly J(R). Maximal ideals are in a sense easier to look for than annihilators of modules. This characterization is deficient, however, because it does not prove useful when working computationally with J(R). The left-right symmetry of these two definitions is remarkable and has various interesting consequences. This symmetry stands in contrast to the lack of symmetry in the socles of R, for it may happen that soc(R_R) is not equal to soc(_RR). If R is a non-commutative ring, J(R) is not necessarily equal to the intersection of all maximal two-sided ideals of R. For instance, if V is a countable direct sum of copies of a field k and R=End(V) (the ring of endomorphisms of V as a k-module), then J(R)=0 because R is known to be von Neumann regular, but there is exactly one maximal double-sided ideal in R consisting of endomorphisms with finite-dimensional image. (Lam 2001, p. 46, Ex. 3.15)

• J(R) equals the sum of all superfluous right ideals (or symmetrically, the sum of all superfluous left ideals) of R. Comparing this with the previous definition, the sum of superfluous right ideals equals the intersection of maximal right ideals. This phenomenon is reflected dually for the right socle of R: soc(R_R) is both the sum of minimal right ideals and the intersection of essential right ideals. In fact, these two astounding relationships hold for the radicals and socles of modules in general.

• As defined in the introduction, J(R) equals the intersection of all annihilators of simple right R-modules, however it is also true that it is the intersection of annihilators of simple left modules. An ideal that is the annihilator of a simple module is known as a primitive ideal, and so a reformulation of this states that the Jacobson radical is the intersection of all primitive ideals. Although this characterization is not useful computationally, or as useful as the previous two characterizations in aiding intuition, it is useful in studying modules over rings. For instance, if U is right R-module, and V is a maximal submodule of U, U·J(R) is contained in V, where U·J(R) denotes all products of elements of J(R) (the "scalars") with elements in U, on the right. This follows from the fact that the quotient module, U/V is simple and hence annihilated by J(R). As another example, this result motivates Nakayama's lemma.

• J(R) is the unique right ideal of R maximal with the property that every element is right quasiregular. Alternatively, one could replace "right" with "left" in the previous sentence. This characterization of the Jacobson radical is useful both computationally and in aiding intuition. Furthermore, this characterization is useful in studying modules over a ring. Nakayama's lemma is perhaps the most well-known instance of this. Although every element of the J(R) is necessarily quasiregular, not every quasiregular element is necessarily a member of J(R).

• While not every quasiregular element is in J(R), it can be shown that y is in J(R) if and only if xy is left quasiregular for all x in R. (Lam 2001, p. 50)

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):

• The notion of left quasiregularity can be generalized in the following way. Call an element a in R left generalized quasiregular if there exists c in R such that c+a-ca= 0. Then J(R) consists of every element a for which ra is left generalized quasiregular for all r in R. It can be checked that this definition coincides with the previous quasiregular definition for rings with unity.
• For a ring without unity, the definition of a left simple module M is amended by adding the condition that R•M ≠ 0. With this understanding, J(R) may be defined as the intersection of all simple left R modules, or just R if there are no simple left R modules. Rings without unity with no simple modules do exist, in which case R=J(R), and the ring is called a radical ring. By using the generalized quasiregular characterization of the radical, it is clear that if one finds a ring with J(R) nonzero, then J(R) is a radical ring when considered as a ring without unity.