Jacobson Radical - Examples

Examples

• Rings for which J(R) is {0} are called semiprimitive rings, or sometimes "Jacobson semisimple rings". The Jacobson radical of any field, any von Neumann regular ring and any left or right primitive ring is {0}. The Jacobson radical of the integers is {0}.
• The Jacobson radical of the ring Z/12Z (see modular arithmetic) is 6Z/12Z, which is the intersection of the maximal ideals 2Z/12Z and 3Z/12Z.
• If K is a field and R is the ring of all upper triangular n-by-n matrices with entries in K, then J(R) consists of all upper triangular matrices with zeros on the main diagonal.
• If K is a field and R = K] is a ring of formal power series, then J(R) consists of those power series whose constant term is zero. More generally: the Jacobson radical of every local ring is the unique maximal ideal of the ring, which consists precisely of the ring's non-units.
• Start with a finite, acyclic quiver Γ and a field K and consider the quiver algebra KΓ (as described in the quiver article). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1.
• The Jacobson radical of a C*-algebra is {0}. This follows from the Gelfand–Naimark theorem and the fact for a C*-algebra, a topologically irreducible *-representation on a Hilbert space is algebraically irreducible, so that its kernel is a primitive ideal in the purely algebraic sense (see spectrum of a C*-algebra).